System For Indirectly Measuring A Geometric Dimension Related To An Opening In An Apertured Exterior Surface of A Part Based On Direct Measurements Of The Part When Fixtured At A Measurement Station

ABSTRACT

A system for indirectly measuring a geometric dimension related to an opening in an apertured exterior surface of a part such as an ammunition case based on direct measurements of the part when fixtured at a measurement station is provided. The system includes first and second holding devices for holding the part therebetween in a part-retaining position in which the part is firmly held between the devices at its end surfaces. In one embodiment, a portion of each of the holding devices extends into its respective opening in the part-retaining position. The system also includes a head apparatus which has a plurality of radiation sources for successively directing arrays of planes of radiation at the holding devices and at the part, and a plurality of receiver modules for measuring the amount of radiation present in unobstructed planar portions of the planes to obtain holding device and part signals. The system further includes a movable stage subsystem coupled to the head apparatus for translating the head apparatus. The system still further includes a signal processor for processing the holding device and part signals to obtain data and a data processor for processing the data to obtain the direct measurements. The data processor determines the geometric dimensions related to the openings based on the direct measurements.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is related to the following commonly-owned U.S. patentapplications which are filed on Sep. 19, 2008:

-   1) Non-contact Method and System for Inspecting Parts (U.S. Ser. No.    12/233,289);-   2) Method and System for Inspecting Parts Utilizing Triangulation    (U.S. Ser. No. 12/233,831); and-   3) Method for Precisely Measuring Position of a Part to Be Inspected    at a Part Inspection Station (U.S. Ser. No. 12/233,821).

This application is related to the following commonly-owned U.S. patentapplications which were filed on Oct. 23, 2007:

-   1) Method and System for Optically Inspecting Parts (U.S. Ser. No.    11/977,117);-   2) Method for Estimating Thread Parameters of a Part (U.S. Ser. No.    11/977,097);-   3) Optical Modules and Method of Precisely Assembling Same (U.S.    Ser. No. 11/977,102);-   4) Method and Inspection Head Apparatus for Optically Measuring    Geometric Dimensions of a Part (U.S. Ser. No. 11/977,010);-   5) Apparatus for Quickly Retaining and Releasing Parts to be    Optically Measured (U.S. Ser. No. 11/977,091);-   6) Calibration Device for Use in an Optical Part Measuring System    (U.S. Ser. No. 11/977,114); and-   7) Method and System for Generating Calibration Data For Use In    Calibrating A Part Inspection System (U.S. Ser. No. 11/975,977).

This application is further related to commonly-owned U.S. patentapplication entitled “Profile Inspection System for Threaded and AxialComponents” filed Aug. 25, 2006 and having U.S. Ser. No. 11/510,402.

BACKGROUND OF THE INVENTION

1. Field of the Invention

This invention relates to systems for indirectly measuring geometricdimensions related to openings in apertured exterior surfaces of partssuch as ammunition cases based on direct measurements of the parts whenfixtured at a measurement station.

2. Background Art

Traditional manual, gauging devices and techniques have been replaced tosome extent by automatic inspection methods and systems. However, suchautomatic inspection methods and systems still have a number ofshortcomings associated with them.

Inspection of defects on small arms ammunition cartridges and cases is avital aspect in the manufacturing process, allowing for maintenance of ahigh level of quality and reliability in the munitions industry.Standards have been developed and applied by manufacturers for manyyears to assist in classifying various types of defects. Alternatively,a military standard is used such as that introduced in 1958 by the USDepartment of Defense, MIL-STD-636. For small arms ammunition calibersup to 0.50, this standard serves to evaluate and illustrate a practicalmajority of defects assembled as a result of extensive surveys coveringall the small arms ammunition manufacturing facilities in the UnitedStates.

U.S. Pat. No. 4,923,066 discloses an automatic visual inspection systemfor small arms ammunition which sorts visual surface flaws at high speedaccording to established standards which can be tailored to fit specificneeds. The system employs techniques for performing inspectionindependently of human inspectors and allows for quick changeovers inthe type of ammunition to which it is applied. The system comprisesinterface apparatus for receiving a supply of ammunition cartridges andproviding each cartridge with a predetermined orientation, conveyingapparatus for locating each of the cartridges for inspection in at leastone inspection station, apparatus for imaging selected areas of eachcartridge to provide video surface feature data associated therewith,and apparatus for processing the video surface feature data to detectthe presence of a predetermined set of characteristics and provideoutput signals in accordance therewith, the conveying apparatus beingoperated to sort each of the inspected cartridges in accordance with theoutput signals. A preferred embodiment comprises four subsystems, afeeding subsystem, an imaging and handling subsystem, an operationsubsystem, and a computer subsystem. The imaging and handling subsystemprovides each cartridge with the necessary orientation for inspection bya video camera feeding video surface feature data to an image processingcomputer. The image processing computer makes a very high speedcomputation based on image processing techniques to decide whether thecartridges have manufacturing defects for sorting purposes. Since manysurface flaws look the same in two dimensions such as scratches andsplits or acid holes and stains, special lighting of the cartridges isused so that discrimination between them can be achieved on the basis ofoff-specular reflections.

U.S. Pat. No. 7,403,872 discloses a method and system for inspectingmanufactured parts such as cartridges and cartridge cases and sortingthe inspected parts. The method and system inspect the manufacturedparts for a wider variety of defects than previously possible. Thesystem includes an illumination assembly for evenly illuminating aplurality of annular, exterior side surfaces of a part when the part islocated in a circumference vision station with rings of strobedradiation to generate corresponding reflected radiation signals. Aplurality of imaging detectors in the form of CCD cameras are located atthe vision station to generate a plurality of side images. The systemfurther includes at least one side image processor for processing theside images of each part to identify parts having an unacceptabledefect. Mouth and primer vision stations are disclosed as well as thecircumference vision station.

WO 2005/022076 discloses a plurality of light line generators (72) whichgenerate associated beams of light (26) that intersect a part (14) to beinspected. Each beam of light (26) illuminates at least one side of thepart (14) with a line of light occluded by the part (14), and at leastthree light responsive sensors (104) provide for generating a signal(24) responsive to an occlusion of a corresponding line of light on acorresponding side of at least one side of the part (14). Each of thelight responsive sensors is responsive to an occlusion at a differentazimuthal location. A processor (28) analyzes the signals (24) inrelation to a measure of relative location of the part (14) from amotion (18) or position sensor. The part (14) may be released from aclamp (52) to drop through the beams of light (26), or the beams oflight (26) may be moved relative to the part (14).

U.S. Pat. No. 6,313,948 discloses an optical beam shaper for productionof a uniform sheet of light for use in a parts inspection system havinga light source including a coherent light generator, a diffractive beamshaper, and lens elements.

U.S. Pat. No. 6,285,034 discloses an inspection system for evaluatingrotationally asymmetric workpieces for conformance to configurationcriteria. The system has a track for causing the workpieces to translatethrough a test section. The test section includes a plurality ofelectromagnetic energy sources. The plurality of electromagnetic energysources are oriented with respect to the track such that the workpiecesocclude the plurality of electromagnetic energy sources upon passingthrough the test section. The test section further has electromagneticenergy detectors for receiving the electromagnetic energy to provideoutput signals related to the intensity of the occluded electromagneticenergy incident on the electromagnetic energy detectors, and a signalprocessor for receiving and processing the output signals.

U.S. Pat. No. 6,252,661 discloses an inspection system for evaluatingworkpieces for conformance to configuration criteria. The systemincludes a track for causing workpieces to translate through a testsection. The test section includes a light source for producing auniform sheet of light. The light source is oriented with respect to thetrack such that the workpieces occlude the uniform sheet of light uponpassing through the test section. The test section further has a videosystem for receiving the occluded uniform sheet of light, providingoutput signals related to the intensity of the occluded uniform sheet oflight incident on the video system, and a signal processor for receivingand processing the output signals.

U.S. Pat. No. 6,959,108 discloses an inspection system whereinworkpieces to be inspected are consecutively and automatically launchedto pass unsupported through the field of view of a plurality of cameras.As a workpiece passes through the field of view of the cameras, a sensoris activated which communicates with a computer system to activate thecameras to capture an unobstructed image, or image data, of theworkpiece. The image data is then analyzed by a computer program toverify whether the image data indicates that the workpiece does not meetestablished criteria and therefore is considered defective. If the imagedoes not meet the established criteria, the workpiece is rejected andsegregated from workpieces which have not been identified as defective.

U.S. Pat. No. 5,608,530 discloses a laser for producing a beam ofradiation which is then refined in cross-sectional dimension by use ofplano-cylindrical lenses. The refined beam of radiation falls incidenton a part to be measured. The unobstructed portion of the beam is thenbifurcated by a pair of reflective surfaces which produce non-parallelradiating beams. Each resulting beam comprises the unobstructed portionof radiation which has passed radially opposed halves of the part. Themagnitude of radiation present in each non-parallel radiating beam isthen measured.

U.S. Pat. No. 4,831,251 discloses an optical device for discriminatingthreaded workpiece by the handedness by their screw thread profiles. Thedevice present a pair of light beams which pass generally tangent to theworkpiece at angularly displaced positions. The light beams are inclinedto follow the helix direction of a given handedness of a workpiece. Uponaxial advancement of a workpiece through the device, a chopped outputfrom the photodetectors indicates that the handedness of the threadsmatches the inclination of the light beams. The oppositely threadedworkpiece, however, provides a generally constant DC output. Withappropriate signal processing electronics, an automatic system fordiscriminating workpieces by thread handedness is provided.

U.S. Pat. No. 5,383,021 discloses a non-contact inspection systemcapable of evaluating spatial form parameters of a workpiece to provideinspection of parts in production. The system causes parts to besequentially loaded onto an inclined track where they pass through atest section. The test section includes a length detection array formeasuring the length of the workpiece, which includes a sourcegenerating a sheet of light oriented in the longitudinal direction ofthe workpiece. The profile of the parts are evaluated by one or morelight sources also creating a sheet of light oriented transversed to thelongitudinal axis of the parts. Single channel photodetectors areprovided for each of the sources which provides an analog output of theextent to which each sheet of light is occluded by the part. Theseoutputs are analyzed through appropriate signal processing hardware andsoftware to generate length and profile data related to the workpiecegeometry.

U.S. Pat. No. 5,568,263 discloses a non-contact inspection systemcapable of evaluating spatial form parameters of a workpiece to provideinspection of parts in production. The system causes parts to besequentially loaded onto an incline track where they pass through a testsection. The test section includes a length detection array formeasuring the length of the workpiece, which includes a sourcegenerating a sheet of light oriented in the longitudinal direction ofthe workpiece. The profile of the parts are evaluated by one or morelight sources also creating a sheet of light oriented transverse to thelongitudinal axis of the parts. First and second pairs of single channelphotodetectors are provided for each of the light sources which providesa pair of analog outputs of the extent to which each sheet of light isoccluded by the part, as well as an ability to eliminate noise orscintillation caused by a point source of light, for example with alaser light source. These outputs are analyzed through appropriatesignal processing hardware and software to generate length and profiledata related to the workpiece geometry.

U.S. Pat. No. 4,852,983 discloses an optical system which simulates theoptical effect of traveling over a large distance on light travelingbetween reference surfaces.

U.S. Patent Application Publication No. 2005/0174567 discloses a systemto determine the presence of cracks in parts. The presence of cracks isdetermined through the use of an imaging device and illumination source.The part is moved along a track where it is sensed by a position sensorto initiate the inspection. The illumination source projects a sheet oflight onto the part to be inspected. The line formed by the intersectionof the sheet of light and the part is focused onto the imaging device.The imaging device creates a digital image which is analyzed todetermine if cracks are present on the part.

U.S. Patent Application Publication No. 2006/0236792 discloses aninspection station for a workpiece including a conveyor, a mechanism forrotating the workpiece, and a probe. The conveyor includes a fixture forlocating the workpiece and the conveyor is configured to translate theworkpiece in a linear manner. A mechanism, such as a belt, engages theworkpiece thereby rotating the workpiece within the fixture. The probeis configured to indicate if the workpiece conforms to quality criteria.To facilitate inspection while the conveyor translates the workpiece,the probe is attached to a stage where the stage is configured to movethe probe synchronously with the workpiece over an inspection region.

U.S. Pat. No. 6,289,600 discloses a non-contact measuring device fordetermining the dimensions of a cylindrical object, such as a pipe. Thedevice may use a programmable industrial robot with an arm terminatingin a rotary joint, or wrist. A non-contact measurement assembly, whichpreferably includes a triangulation laser, mounted to the robot's wrist.The wrist includes a binary wrist encoder for providing the angularposition of the wrist to an attached computer. Likewise, the laserprovides a signal representative of the distance from the laser to thesurface of the pipe to the computer. The robot positions the measurementassembly in a predetermined location within the pipe and rotates themeasurement assembly. The computer correlates the measurements from thelasers with the angular position at which they were taken, and itanalyzes this data to determine the dimensions of the pipe, includingthe true profile of the surface and cross-sectional shape.

U.S. Pat. No. 5,521,707 discloses a non-contact laser-based sensorguided by a precision mechanical system scans a thread form producing aset of digitized images of the thread form. The digitized images areanalyzed to derive quantitative information about thread characteristicssuch as pitch, lead, root radius, flank angle, surface roughness, helixvariation, and pitch diameter. Thread form data may be stored and laterretrieved in order to provide traceability and verification of threadform measurements to a predetermined thread form specification. Theapparatus and method of the invention can measure characteristics forshapes similar to threads, such as gears and dies used for cold formingprocesses. The thread measurement system can be mounted on thread makingmachines and used for on-line monitoring and control of the threadforming process.

U.S. Pat. No. 4,547,674 discloses a method and apparatus for inspectinggear geometry via optical triangulation. Preferred machine constructionuses specialized geometrical relationships of sensor to gear, andspecialized high accuracy photodetector array based laser triangulationsensor units of large range and high response speed.

U.S. Pat. No. 4,970,401 discloses a non-contact triangulation probesystem including a base plate and a first non-contact triangulationprobe including a light source mounted on a first movable slide. Thefirst slide is disposed on the base plate and arranged for slidingmotion therealong, a source of movement for the first slide beingassociated therewith. A first position sensor determines the relativeposition of the first probe and provides an output signal correspondingthereto. A second non-contact triangulation probe includes a lightsource mounted on a second movable slide. The second slide is disposedon the base plate and arranged for sliding motion in the same directionas the first slide, a source of movement for the second slide beingassociated therewith. A second position sensor determines the relativeposition of the second probe and provide an output signal correspondingthereto. The first and second probes are arranged to face each othersuch that the light beams emitted by the probes coincide on the sameoptical axis. A microcomputer is arranged to actuate the source ofmovement for the first and second slides, to receive the output signalsfrom the first and second position sensors, and to receive outputsignals from the first and second probes.

U.S. Pat. Nos. 5,168,458 and 5,170,306 disclose methods and systems forgaging threaded fasteners to obtain trilobular parameters.

Other U.S. patents related to the invention include: U.S. Pat. Nos.4,315,688; 4,598,998; 4,644,394; 4,852,983; 4,906,098; 5,521,707;5,646,724; 5,291,272; 6,055,329; 4,983,043; 3,924,953; 5,164,995;4,721,388; 4,969,746; and 5,012,117.

While the systems described in the above-noted commonly-owned U.S.patent applications provide an excellent tool for first articleinspection, random sampling QA and for generating real-time processcontrol feedback, there are several important features on parts such asammunition cartridge cases (empty brass) which cannot be measured usingthe hardware configuration described in these systems. These features ofinterest include Mouth Inner Diameter, Mouth Wall Thickness, PrimerPocket Diameter and Primer Pocket Depth when the parts are cartridgecases.

SUMMARY OF THE INVENTION

An object of the present invention is to provide an improved system forindirectly measuring a geometric dimension related to an opening in anapertured exterior surface of a part such as an ammunition case based ondirect measurements of the part when fixtured at a measurement station.

In carrying out the above object and other objects of the presentinvention, a system for indirectly measuring a geometric dimensionrelated to an opening in an apertured exterior surface of a part basedon direct measurements of the part when fixtured at a measurementstation is provided. The system includes first and second holdingdevices for holding a part therebetween in a part-retaining position inwhich the part is firmly held between the devices at the aperturedexterior surface. A portion of one of the holding devices extends intothe opening in the part-retaining position. A head apparatus includes aplurality of radiation sources for successively directing a first arrayof planes of radiation at the one of the holding devices and a secondarray of planes of radiation at the part in the part-retaining positionso that the one of the holding devices and the part occlude theirrespective planes of radiation to create corresponding arrays ofunobstructed planar portions of the planes of radiation. Each of theunobstructed planar portions contains an amount of radiation which isrepresentative of a respective external geometric dimension of eitherthe one of the holding devices or the part. The head apparatus furtherincludes a plurality of receiver modules for measuring the amount ofradiation present in each of the unobstructed planar portions to obtainholding device and part signals. The system still further includes amovable stage subsystem coupled to the head apparatus for translatingthe head apparatus relative to the devices and the part so that theplanes of radiation scan the one of the holding devices and the part inthe part-retaining position. The system further includes a signalprocessor for processing the holding device and part signals to obtaindata and a data processor for processing the data to obtain the directmeasurements. The data processor determines the geometric dimensionrelated to the opening based on the direct measurements.

The holding devices may be movable relative to each other between thepart-retaining position and a part-release position in which the devicesrelease the part for removal of the part from the station.

The portion of the one of the holding devices may have a sloped regionat which the portion seats on the opening at the exterior surface. Theportion may have a cross sectional shape substantially identical to theshape of the opening at the exterior surface. The sloped region maydefine a range of sizes of the opening at the exterior surface that canbe indirectly measured at the station.

The portion of the one of the holding devices may be frustum-shaped. Thegeometric dimension may be distance across the opening at the exteriorsurface.

The geometric dimension related to the opening may be thickness of awall which defines the opening at the exterior surface.

The one of the holding devices may include a spring for spring-loadingthe portion so that the portion applies a variable clamping force to thepart.

The sloped region may have a constant slope.

The system may include a memory coupled to the data processor to storedata which represents height of the portion. The first holding devicemay include the portion. The portion may have a distal end surface whichengages an inner end surface of the opening in the part-retainingposition. The data processor may process the data including the storeddata to obtain depth of the opening in the part.

The first holding device may include first and second portions whichextend into the opening in the part-retaining position. The firstportion may have a sloped region and a cross sectional shapesubstantially identical to the shape of the opening at the exteriorsurface. The sloped region may define a range of sizes of the opening atthe exterior surface that can be indirectly measured at the station. Thesecond portion may have a distal end surface which engages an inner endsurface of the opening in the part-retaining position.

The first and second portions may be movable relative to each other. Thefirst holding device may further include a spring for spring loading thefirst portion so that the first portion can apply a variable clampingforce to the part.

The part may be an ammunition case.

Further in carrying out the above object and other objects of thepresent invention, a system for indirectly measuring geometricdimensions related to openings in apertured exterior surfaces of a partbased on direct measurements of the part when fixtured at a measurementstation is provided. The system includes first and second holdingdevices for holding a part therebetween in a part-retaining position inwhich the part is firmly held between the devices at the aperturedexterior surfaces of the part. A portion of each of the holding devicesextends into its respective opening in the part-retaining position. Ahead apparatus includes a plurality of radiation sources forsuccessively directing a first array of planes of radiation at the firstholding device, a second array of planes of radiation at the part and athird array of planes of radiation at the second holding device in thepart-retaining position so that the holding devices and the part occludetheir respective planes of radiation to create corresponding arrays ofunobstructed planar portions of the planes of radiation. Each of theunobstructed planar portions contains an amount of radiation which isrepresentative of a respective external geometric dimension of one ofthe holding devices or the part. The head apparatus further includes aplurality of receiver modules for measuring the amount of radiationpresent in each of the unobstructed planar portions to obtain holdingdevice and part signals. The system still further includes a movablestage subsystem coupled to the head apparatus for translating the headapparatus relative to the holding devices and the part so that theplanes of radiation scan the holding devices and the part in thepart-retaining position. The system further includes a signal processorfor processing the holding device and part signals to obtain data and adata processor for processing the data to obtain the directmeasurements. The data processor determines the geometric dimensionsrelated to the openings based on the direct measurements.

The holding devices may be movable relative to each other between thepart-retaining position and a part-release position in which the devicesrelease the part for removal of the part from the station.

The portion of each of the holding devices which extends into itsrespective opening may have a sloped region at which the portions seaton their respective openings at their exterior surfaces. Each of theportions may have a cross sectional shape substantially identical to theshape of its respective opening at its exterior surface. Each slopedregion may define a range of sizes of its respective opening at itsexterior surface that can be indirectly measured at the station.

The portion of each of the devices may be frustum-shaped. Each geometricdimension may be distance across its respective opening at its exteriorsurfaces.

One of the geometric dimensions related to the openings may be thicknessof a wall which defines one of the openings at its exterior surface.

Each of the holding devices may include a spring for spring-loading itsrespective portion so that the portions of the devices apply a variableclamping force to the part.

Each of the sloped regions may have a constant slope.

The system may include a memory coupled to the data processor to storedata which represents height of the portion of the first holding device.The portion of the first holding device may have a distal end surfacewhich engages an inner end surface of its opening in the part-retainingposition. The data processor may process the data including the storeddata to obtain depth of its opening in the part.

The first holding device may include first and second portions whichextend into their respective opening in the part-retaining position. Thefirst portion may have a sloped region and a cross sectional shapesubstantially identical to the shape of its opening at its exteriorsurface. The sloped region may define a range of sizes of its opening atits exterior surface that can be indirectly measured at the station. Thesecond portion may have a distal end surface which engages an inner endsurface of its opening in the part-retaining position.

The first and second portions may be movable relative to each other. Thefirst holding device may further include a spring for spring loading thefirst portion so that the first portion can apply a variable clampingforce to the part.

The part may be an ammunition case.

Still further in carrying out the above object and other objects of thepresent invention, a system for indirectly measuring geometricdimensions related to openings in apertured end surfaces of anammunition case based on direct measurements of the case when fixturedat a measurement station is provided. One end surface of the case islocated at a mouth end of the case and the other end surface is locatedat a primer end of the case. The system includes first and secondholding devices for holding the case therebetween in a case-retainingposition in which the case is firmly held between the devices at the endsurfaces of the case. A portion of each of the holding devices extendsinto its respective opening in the case-retaining position. A headapparatus includes a plurality of radiation sources for successivelydirecting a first array of planes of radiation at the first holdingdevice, a second array of planes of radiation at the case and a thirdarray of planes of radiation at the second holding device in thecase-retaining position so that the holding devices and the case occludetheir respective planes of radiation to create corresponding arrays ofunobstructed planar portions of the planes of radiation. Each of theunobstructed planar portions contains an amount of radiation which isrepresentative of a respective external geometric dimension of one ofthe holding devices or the case. The head apparatus further includes aplurality of receiver modules for measuring the amount of radiationpresent in each of the unobstructed planar portions to obtain holdingdevice and case signals. The system further includes a movable stagesubsystem coupled to the head apparatus for translating the headapparatus relative to the holding devices and the case so that theplanes of radiation scan the holding devices and the case in thecase-retaining position. The system still further includes a signalprocessor for processing the holding device and case signals to obtaindata and a data processor for processing the data to obtain the directmeasurements. The data processor determines geometric dimensions relatedto the openings based on the direct measurements.

The holding devices may be movable relative to each other between thecase-retaining position and a case-release position in which the devicesrelease the case for removal of the case from the station.

The portion of each of the holding devices which extends into itsrespective opening may have a sloped region at which the portions seaton their respective openings at their exterior surfaces. Each of theportions may have a cross sectional shape substantially identical to theshape of its respective opening at its exterior surface. Each slopedregion may define a range of sizes of its respective opening at itsexterior surface that can be indirectly measured at the station.

The portion of each of the devices may be frustum-shaped. The geometricdimensions may be inner diameter located at the mouth end and primerpocket diameter located at the primer end.

One of the geometric dimensions related to the openings may be thicknessof a wall which defines the opening at the mouth end.

Each of the holding devices may include a spring for spring-loading itsrespective portion so that the portions of the devices apply a variableclamping force to the case.

Each of the sloped regions may have a constant slope.

The system may further include a memory coupled to the data processor tostore data which represents height of the portion of the first holdingdevice. The portion of the first holding device may have a distal endsurface which engages an inner end surface of a primer pocket located atthe primer end in the case-retaining position. The data processor mayprocess the data including the stored data to obtain primer pocketdepth.

The first holding device may include first and second portions whichextend into their respective opening in the case-retaining position. Thefirst portion may have a sloped region and a cross sectional shapesubstantially identical to the shape of its opening at its end surface.The sloped region may define a range of sizes of the opening at theprimer end that can be indirectly measured at the station. The secondportion may have a distal end surface which engages an inner end surfaceof a primer pocket located at the primer end in the case-retainingposition.

The first and second portions may be movable relative to each other. Thefirst holding device may further include a spring for spring loading thefirst portion so that the first portion can apply a variable clampingforce to the case.

The ammunition case may be a cartridge case.

The above object and other objects, features, and advantages of thepresent invention are readily apparent from the following detaileddescription of the best mode for carrying out the invention when takenin connection with the accompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a schematic perspective view of a part inspection systemincluding measurement hardware;

FIG. 2 is a schematic, side elevational view of a part holder base (witha side plate of a container removed) and an upper tooling unit with apart held between colinear upper and lower holding devices and acalibration fixture or device mounted colinear with the part;

FIG. 3 a is a perspective schematic view of the part holder basesupporting the lower holding device and the calibration device suspendedfrom it;

FIG. 3 b is a top plan view of the base of FIG. 3 a;

FIG. 3 c is a sectional view taken along lines 3 c-3 c of FIG. 3 b andfurther including a part support subsystem including the lower holdingdevice;

FIG. 4 a is a side elevational view of the calibration device or cone ofFIG. 3 a;

FIG. 4 b is a top plan view of the cone of FIG. 4 a;

FIG. 4 c is a sectional view taken along lines 4 c-4 c of FIG. 4 b;

FIG. 5 is a top plan view of an optical head with its top cover plateremoved in order to provide an interior view of the optical head;

FIG. 6 is a schematic perspective view of a safety enclosure which mayenclose the basic measurement hardware of FIG. 1;

FIG. 7 is a side elevational view, partially in cross-section, of theupper tooling unit including the upper holding device of FIG. 2;

FIG. 8 is a schematic block diagram which illustrates basic beam linesubsystem components wherein a laser generates a laser beam, a mirrorreflects the laser light beam, and a light plane generator modulegenerates a laser light plane which is directed or projected onto apart; it is to be understood that in addition to a part, such as anammunition case, laser light planes are also directed or projected ontothe upper and lower holding device in at least one embodiment of thepresent invention;

FIG. 9 is a schematic perspective view of a cylindrical part which isintersected by a projected laser light plane; it is to be understoodthat in addition to a part the upper and lower holding devices are alsointersected by projected laser light planes;

FIG. 10 is a schematic view, partially broken away, of various positionmeasurement system components together with the optical head which ismounted on a stage to move with it just prior to scanning thecalibration device;

FIG. 11 are graphs of various raw sensor signals generated when thebeginning edge of the calibration cone is scanned;

FIG. 12 is a schematic block diagram of various laser transmitters andreceivers contained within the optical head;

FIG. 13 is a top plan schematic view of one of a part, an upper holdingdevice and a lower holding device illuminated by multiple planes oflaser light with various tangential shadow rays and points;

FIG. 14 is a graph of sensor height data created with a threaded partsuch as a standard thread plug go gage;

FIG. 15 is a symbolic sketch of the calibration cone's profile withvarious usage regions;

FIG. 16 is a schematic block diagram which illustrates raw sensor dataprocessed with calibration data to obtain calibrated sensor data;

FIG. 17 is a thread model superimposed on a graph which illustratesrough thread peaks, troughs (or more correctly “crests” and “roots”) andcrossings obtained by the intersection of a pitch diameter line withflank lines;

FIG. 18 is a graph similar to the graph of FIG. 17 in which selectedthread concepts are illustrated;

FIG. 19 is a single thread form that illustrates thread flank lines, athread flank line data extraction region and a wire position;

FIG. 20 is a schematic diagram that illustrates a 3-point distancemeasurement between a reference line defined by a pair of virtual wirecenters on one side of a threaded part and a single wire center on theopposite side of the part;

FIG. 21 is a screen shot of a graph of sensor height data with virtualwires in roots and with intermediate data illustrated;

FIG. 22 is a screen shot which illustrates an enlarged thread pitch withintermediate data;

FIG. 23 is a top plan schematic view of a cylindrical part with a beamof light deflected at an angle θ_(refl)=2θ by a perfectly reflectingsurface of the part;

FIG. 24 is a top plan schematic view of one of a part, an upper holdingdevice and a lower holding device which scatters light from a plane oflight which scattered light is blocked by light plane receiver apertureslits;

FIG. 25 illustrates a conical shape (i.e., frustum) utilized in thecalibration fixture;

FIG. 26 is a sketch, partially broken away, similar to the sketch ofFIG. 15 which shows a schematic outline of the sensor signal produced bythe calibration cone and various support structures;

FIG. 27 is a sketch that illustrates a full open signal level computedfrom data in the full open estimation region;

FIG. 28 is a graph that shows the data regions of FIG. 27;

FIG. 29 is a sketch that illustrates cone projection geometry;

FIG. 30 is a sketch of α₁,α₂,α₃,α₄ which are the projections of {rightarrow over (α)} on the y′-axis for the θ=22.5, 67.5, 112.5, 157.5 degreelaser sensor systems; the positive y′-axis is the right sensor and thenegative y′-axis is the left sensor direction;

FIG. 31 is a schematic perspective view of an adjustment fixture forassembling optical modules;

FIG. 32 is a graph illustrating a relatively non-level laser beamprofile;

FIG. 33 is a graph illustrating a relatively flat laser beam profile;

FIG. 34 is a graph illustrating a relatively collimated laser beamprofile;

FIG. 35 is a graph illustrating a distorted, generally rectangular laserbeam profile;

FIG. 36 is a graph illustrating a non-distorted, generally rectangularlaser beam profile;

FIG. 37 is an exploded perspective view of a typical transmitter moduleincluding its transmitter mount and its supported set of opticalcomponents;

FIG. 38 is a back view of the transmitter mount of FIG. 37;

FIG. 39 is a sectional view of the transmitter mount taken along lines39-39 of FIG. 38;

FIG. 40 is a top plan view of the transmitter mount;

FIG. 41 is a schematic perspective view of the assembled transmittermodule;

FIG. 42 is a front view of the assembled transmitter module;

FIG. 43 is a sectional view of the transmitter module taken along lines43-43 of FIG. 42;

FIG. 44 is a top plan view of the assembled transmitter module;

FIG. 45 is an exploded perspective view of a laser beam steering mirrorassembly;

FIGS. 46 a-46 d are top plan views of the assembly of FIG. 45 and havingdifferent angular portions for use in the optical head of FIG. 5;

FIG. 47 is an exploded perspective view of a typical receiver moduleincluding its receiver mount and its supported set of opticalcomponents;

FIG. 48 is a top view of a shadow of a part on left and right sensors;

FIG. 49 is a top perspective view of a 3-D coordinate system whichprovides a definition of “tilt angle”;

FIG. 50 is a top perspective view of a scanned part having a pair ofpart slices spaced apart over a measurement interval;

FIG. 51 is a front perspective view of cartridge case including anapertured top surface located at a mouth end of the case;

FIG. 52 is a bottom perspective view of an apertured bottom surfacelocated at a primer end of the case;

FIG. 53 is a side perspective view of an upper holding deviceconstructed in accordance with one embodiment of the present invention;

FIG. 54 is a side sectional view of the device of FIG. 53 taken alonglines 54-54;

FIG. 55 is a side perspective view of a lower holding device alsoconstructed in accordance with one embodiment of the present invention;

FIG. 56 is a side perspective view of an upper holding deviceconstructed in accordance with another embodiment of the presentinvention; and

FIG. 57 is a side sectional view of the device of FIG. 56 taken alonglines 57-57.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

The overall system described in this application is referred to as“Laser Lab.” Laser Lab is a trademark of the assignee of thisapplication. It is to be understood that numerous inventions aredescribed in this application, of which only some are claimed. The otherdisclosed inventions are claimed in the applications noted in the CrossReference to Related Applications part of this application.

The LaserLab described in the above-noted commonly owned applicationsdoes at least one thing very well: quickly and accurately measure allexternal characteristics on a manufactured cylindrical part. The goal ofat least one embodiment of this invention was to find a way to use theinformation available from the LaserLab to learn something about alimited number of specific internal characteristics. While this is notpossible directly, a way was devised to use simple tooling that servesto both fixture the part in the machine or system and, by virtue of thetooling's shape, allow indirect measurement of the characteristics ofinterest. This tooling does not allow measurement of thesecharacteristics directly. Furthermore, the tooling does not include anyadditional sensors. Rather, the tooling allows the existing sensors onthe LaserLab to extract more information from the data it is alreadycapable of collecting, and it does this without any significantadditional effort on the part of the operator.

It is also to be understood that a number of words and phrases areexplained in a Glossary portion of this application. The Glossaryexplains but does not unduly limit the words and phrases contained inthis application.

Laser Lab—Physical Overview

The Laser Lab system (i.e., FIG. 1) includes several physical subsystemsor units.

A PC tower unit (i.e., FIG. 8) contains a computer and a number ofadditional control electronics modules. It has a rear panel withconnectors for light curtain/safety electronics, motor control, linearencoder, measurement signals, and optical head control. The PC towerunit hosts the application program which presents a user interface to anoperator of the Laser Lab system.

Although at least one embodiment of the present invention was motivatedby the needs of the ammunition industry, there are other examples ofmanufactured parts, such as cylindrical parts, which can take advantageof the present invention. For example, both the battery canister andaerosol container industries can utilize at least one embodiment of theinvention. Throughout the discussions below, it is assumed that the partbeing measured is round, and the tooling that interfaces with this partare therefore described as being conical. If the part contains a recesswith a different shape (hexagonal for example), a tooling set with adifferent cross-section can be used. The concept of the invention isotherwise unchanged.

Part holder and upper tooling units (FIGS. 1, 2 and 7) secure or receiveand retain a part, such as an ammunition case, in place for measurement.The upper tooling unit includes a stainless steel rod with a springloaded upper holding device that can move up and down to accommodate awide variety of part sizes. The part holder unit has a base to support alower holding device and its held part or unit under test (UUT) and acalibration device (i.e., FIGS. 3 a-4 c). The calibration cone or deviceis used to measure the relationship between a light sensor output andthe physical measurements represented by the outline dimensions of thecone. The calibration cone or device is not strictly speaking a cone butrather includes a number of frustums (i.e., portions of cones) andcylinders.

An optical head (i.e., FIGS. 1, 5 and 10) is a sealed system containinga number of components including optical measurement components (i.e.,FIGS. 5 and 12). A set of four laser beam lines (one of which is shownin FIG. 8) generate and measure four planes of light.

A slide/base unit (i.e., FIGS. 1 and 10) moves the optical headvertically up and down to make part and holding device measurements. Onevery scan the optical head's eight sensors measure shadow images of thecalibration cone, the part (UUT) (i.e., FIGS. 8, 9 and 13) and theholding devices. Each complete scan thus contains both calibration dataand measurement data, yielding a system that is especially unaffected bytime variations in measurement conditions.

Referring again to the drawing figures, FIG. 1 is a schematicperspective view of the Laser Lab system, generally indicated at 10,including the basic measurement hardware of the system 10. Shown are theoptical head, generally indicated at 12, the part holder/upper toolingunits, generally indicated at 14 and 16, respectively, and thebase/slide unit, generally indicated at 18, which, as also shown in FIG.10, includes a motor 20 coupled to a lead screw 22 which, in turn, iscoupled to a saddle 24 slidably supported by a bearing. The saddle 24 iscoupled to the optical head 12 to move the optical head 12 linearlyalong a vertical stage axis 28 (i.e., FIG. 9). Movement of the stage issensed by a linear encoder 30 which will be described in greater detailbelow.

FIG. 2 is a schematic, side elevational view of the part holder andupper tooling units 14 and 16, respectively. The upper tooling unit 16includes a rod 32 which is manually movable along a central axis of therod 32 in up and down directions by an operator of the system 10. Asshown in greater detail in FIGS. 7 and 51-57, the upper tooling unit 16also includes a spring-loaded upper holding device or tooling assembly,generally indicated at 600, having a tip portion 602 which retains apart such as an ammunition case, generally indicated at 604, to beinspected at an upper apertured end surface 606 by extending into anopening 608 in the case 604. The tip portion 602 is frustum-shaped andhas a sloped region and a cross-sectional shape substantially identicalto the shape of opening 608 in the end surface 606 at a mouth end of thecase 604. The sloped region of the tip portion 602 defines a range ofsizes of the opening 608 that can be measured at the measurementstation.

The holding device 600 has a first central bore 610 formed in acylindrical housing portion 611 and in which a cylindrical, hollowbearing 612 is disposed. The holding device 600 also has a secondcentral bore 614 concentric with the first central bore 610 and in whicha compression spring 616 is disposed within the hollow bearing 612. Ahole 618 extends through an upper portion of the housing portion 611 toreceive a dowel (619 in FIG. 7) by which the holding device 600 issecured to the rod 32. The lower end of the rod 32 has an elongated slot620 extending therethrough which receives and retains the dowel 619. Thelower end of the rod 32 is received within the bore 614 and engages thetop end of the spring 616 to compress the spring 616 within the bore614.

A threaded hole 622 is typically provided at a distal end of the tipportion 602 to threadably receive and retain one of a number of possibletips (not shown) that can contact a surface of a part or case to beinspected.

FIGS. 56 and 57 illustrate a second embodiment of an upper holdingdevice, generally indicated at 600′. Parts of the upper holding device600′ which have the same or similar structure or function as the partsof the first embodiment (i.e., upper holding device 600) have the samereference number but have a prime designation. The upper holding device600′ generally is provided at the lower end of the rod 32 when measuringan opening in an ammunition case which is larger than opening that canbe measured with the holding device 600.

Also illustrated in FIG. 2 are a calibration cone or device, generallyindicated at 40, and a base or top plate 42 of the part holder/base unit14. FIG. 3 a is a perspective schematic view of the part holder baseunit 14. FIGS. 3 a-3 c show the calibration cone 40 and a part holderassembly including a support 44, a collar 46 and a holder 47 into whicha hexagonal base 630 of a lower holding device or tooling assembly,generally indicated at 632, is inserted. The lower holding device 632includes a pin 634 which extends upwardly from a top surface 636 of thebase 630.

Much like the way the upper holding device 600 is secured to the rod 32,the pin 634 has an elongated aperture 638 which receives a dowel pin 640which extends therethrough and through a hole 642 which extends througha cylindrical portion 644 of the lower holding device 632. A compressionspring 646 extends between the top surface 636 and the lower surface ofthe cylindrical portion 644 to bias the cylindrical portion 644 awayfrom the base 630.

A distal end portion 650 of the pin 634 extends through a central bore652 which extends through a tip portion 654 of the lower holding device632. The tip portion 654 is frustum-shaped and has a sloped region and across-sectional shape substantially identical to the shape of an opening656 in an end surface 657 at a primer end of the case 604. The slopedregion of the tip portion 654 defines a range of sizes of the opening656 that can be measured at the measurement station. The distal end 650of the pin 634 moves within the bore 652 after the sloped region of thetip portion 654 extends into the opening 656 of the case 604 and engagesthe side walls which define the opening 656. After engagement, furthermovement of the tip portion 654 downward against the biasing action ofthe spring 646 exposes the distal end 650 of the tip portion 650 whichengages an inner end surface 658 of the opening 656. During thismovement, the dowel pin 640 slides within the aperture 638 of the pin634. As described herein, the depth of the primer pocket or opening 656is determined after scanning the case 604 and the holding device 632.The depth that the pin 634 extends into the opening 656 and engages theend surface 658 is the depth of the opening 656.

In other words, the upper tooling incorporates a shallow cone (orfrustrum) which lowers down inside the mouth of the case and seats onthe inner diameter of the mouth. This cone assembly is spring-loadedwhich provides a small preload to hold the part in place on the stage ofthe system. By defining a region that encompasses the point or line ofcontact between the cone and the case and then measuring the minimumdiameter in that region, the system indirectly measures the innerdiameter (ID) of the mouth.

If the ID on the part to be measured has a radius, the cone will seatdeeper inside the mouth and the diameter measured by the LaserLab willbe larger than the actual ID of the part. In order to minimize thiseffect, the angle of the cone used in this fixture should be as shallowas possible. A cone with a shallow angle results in more accuratemeasurements.

Practically, a cone with a very shallow angle will tend to wedge itselfinside the mouth of the part. The force of the spring in the uppertooling assembly can cause the cone to become stuck inside the casewhich can make unloading parts from the system challenging. Furthermore,a very shallow cone angle can be used only on parts with a very narrowrange of mouth IDs. A cone with a steep angle is less likely to getstuck inside the mouth of the part and is usable on a wider variety ofpart sizes.

In light of these considerations, a relatively shallow cone angle of 10°is preferred with two different cone designs, one for small caliberammunition (50 cal and smaller) and one for medium caliber ammunition(20 mm and larger). This is a convenient break point because it is rarethat a single company manufactures both small and medium caliberammunition.

The wall thickness is calculated by subtracting the directly measured ODfrom the indirectly measured ID and dividing this result by two.

The lower tooling uses a similar concept, with the exception that thistooling configuration also allows the depth of the recess (primerpocket) to be determined. The basis of this assembly is a small postwho's height has been pre-measured by the LaserLab and is stored inmemory of the system. Surrounding this post is a cone which slides upand down on the post. This cone is constrained from coming off the postwith a through-pin which rides in a slot cut into the post. This cone isbiased from underneath by a spring.

When an ammunition case with a primer pocket is placed into theLaserLab, the ID of the primer pocket seats on the angled surface of thecone. The upper tooling presses the case down which, being seated on thecone, causes the spring to compress until the base of the primer pocketcomes to rest on the top of the tooling post.

When the part is scanned in this configuration, the primer pocketdiameter is determined by measuring the minimum diameter at the pointwhere the cone contacts the head of the case in the same manner themouth ID is indirectly measured as described above. The depth of theprimer pocket is measured by measuring the height of the case head(above the base of the post) and subtracting this from the known heightof the tooling post.

The choice of angle for the lower tooling cone (i.e., frustrum) requiresconsideration of all the same factors as the upper tooling cone. Oneadditional detail is the available depth of the part recess. It isdesirable to design this lower tooling assembly so that it can be usedto measure a wide variety of recess depths and diameters. A shallow coneangle severely limits the range of recess aspect ratios (diameter vsdepth) that can be measured with a single tooling design. In order toaccommodate most of the small caliber range, a compromise angle of 18°is preferred. This allows measurement of common primer pockets machinedinto cases sized 7.62 mm and smaller.

A cap 49 covers the assembly. FIGS. 3 a-3 c also show various mechanicalpieces or parts including a mounting plate 50, a riser 52, a bottomplate 54, the top plate 42, an adapter plate 58 for suspending the cone40 at the lower surface of the top plate 42 and a pair of posts 56 forsupporting the top plate 42 at a spaced position above the bottom plate54.

FIG. 4 a is a side elevational view of the calibration cone 40. Asdescribed in detail below, the calibration cone 40 has a preciselymanufactured shape that is utilized in measuring the relationshipbetween raw digitized sensor signals and calibrated physical dimensions.Typically, the cone 40 is sent to a certified laboratory which inspectsthe cone 40. The laboratory then provides a long form certification thatis traceable to NIST.

FIG. 4 b is a top plan view of the cone 40 of FIG. 4 a and FIG. 4 c is aside sectional view of the cone 40 taken along lines 4 c-4 c of FIG. 4b.

FIG. 5 is a top plan view of the optical head 12 with its top coverplate (60 in FIG. 1) removed in order to provide an interior view of theoptical head 12. The head 12 is attached to the saddle 24. Shownadjustably mounted on the bottom base plate 61 of the head 12 are lasersteering mirrors 62, four lasers 64 for generating laser beams,preferably having a wavelength of 650 nm, light plane generator ortransmitter modules, generally indicated at 66, for converting thegenerated laser beams into corresponding light planes, and light planereceiver modules, generally indicated at 68.

The calibration cone has a central hole, preferably about 1.5876 mm(i.e., 1/16″), that is utilized extensively during assembly of thecone/part holder sub-assembly to the slide base unit as illustrated inFIG. 4 c.

At that time, a thin rod (“calibration fine centering rod”) is insertedthrough the cone's central hole when the optical head is in the “down”position. The rod defines the center of the calibration cone moreprecisely than the constant diameter region-O which has a 3.175 mm(i.e., ⅛″) diameter. When the head is in the “down” position the lightplanes pass below the cone, nominally unblocked.

To use the rod, the part holder (44 in FIG. 3 c) is removed from thecone/part holder assembly. Then the rod can be inserted from the top,through the plate 42, and through the cone 40, until it touches thebottom plate 54. The rod is long enough that when it is fully inserted,touching bottom plate 54, a length extends above plate 42, sufficient tomanipulate the rod by hand, moving it up or down.

By moving the “calibration fine centering rod” into and out of the lightplanes, it can be determined if the light plane center line passesthrough the rod. The laser sensor outputs are observed as the rod movesinto and out of the beam. When both left and right sensor outputs show aslight reduction from the effect of the rod blocking the light beam,then the light beam center line passes through the centering rod.

The calibration cone is manufactured with special instructions tofabricate the central hole so that there is a slip fit of thecalibration fine centering rod in the calibration cone's central hole.One rod is paired with each calibration cone, by the cone fabricator.

The use of the calibration fine centering rod makes it possible tomeasure whether or not the cone/part holder center is aligned with theoptical head's four-beam intersection point.

To move the cone/part holder with high precision and complete thealignment/centering operation, four fine-pitch pusher screws areutilized. Each pusher screw is mounted to the triangular base of theslide/base unit (FIG. 1). There is one pusher screw for each of the fourdirections of movement of the cone/part holder plate 50 on thetriangular base, North, East, South, and West.

After the cone/part holder assembly is centered to the optical head beamcenter, the four pusher screws are tightened, maintaining the base plate50 position via compression. Then the pusher screw locking sleeves aretightened. Then the base plate 50 hold down screws are tightened. Thepusher screw locking sleeves can also be secured with an appropriateglue. The above-noted alignment process centers the calibration coneprecisely, with respect to the beam center of the optical head.

FIG. 6 is a schematic perspective view of a safety enclosure, generallyindicated at 70, that encloses the Laser Lab basic measurement hardwareof FIG. 1 which sits inside the enclosure 70. A light curtain isgenerated by a light curtain transmitter 72 and is received by a lightcurrent receiver 74 to guard a physical access opening 76 to the LaserLab. The enclosure 70 and its hardware monitor circuitry (not shown)guarantee that the moving optical head 12 will be stopped before a usercan physically make contact with it.

FIG. 7 is a side elevational view, partially in cross-section, of theupper tooling unit 16. The long rod 32 of the tooling unit 16 can bemoved up/down by almost 9″ (for example) to accommodate a wide range ofUUT (i.e., part and part holder) sizes when an operator manually grips arod handle 78 mounted at a proximal end 80 of the rod 32. The operatorcan release an upper or a lower friction release clamp 82 and 84,respectively, of the tooling unit 16. Support brackets 86 of a supportstructure, generally indicated at 88, hold the upper tooling unit 16 tothe base/slide unit 18. Rod guides 90 supported by a guide support 92which, in turn, is supported by the support brackets 86 and plates 94,hold the rod 32 precisely and also allow it to slide smoothly up/down.Raising the top release clamp 82 allows the rod 32 to move down, whilelowering the bottom release clamp 84 allows the rod 32 to move up.

Laser Lab—Basic Measurements Overview

The Laser Lab is a system for measuring the dimensions of a variety ofmanufactured parts and/or assembled parts such as ammunition casesmanufactured in the munitions industry. These parts may be formed fromcylindrical stock by roll and impact die forming methods or by cuttingwith lathes. The final part can have forms that are built up from basicshape units, such as circular or tapered cylinder, threaded cylinder, oradditional simple shapes such as Trilobe cylinder or hex cylinder.

Substantially all of the measurements obtained with the Laser Lab systemare based on two basic components: (1) the height of a surface from thelight plane split line, and (2) the positions along the optical head'sstage axis corresponding to the various heights.

In addition, the Laser Lab system performs multiple measurements fromfour different measurement directions. However, it is to be understoodthat, depending on the part, the measurements can be taken from as fewas two measurement directions. In some cases, as many as fivemeasurement directions may be needed. This capability of being able toobtain multiple measurements from multiple directions allows the LaserLab to explicitly utilize 3-D shape information, especially in themeasurement of threaded cylinders and non-cylindrical shapes.

Surface Height Relative to the Measurement Axis

In this section the measurements made by a single laser are describedwith reference to FIG. 8. The subsystem that makes these measurements iscalled a beam line. Four beam lines are contained in one optical head asshown in FIG. 5.

The light plane generator module 66 of FIG. 8 creates a light plane froma single laser beam generated by the laser 64. The part creates shadowedand unshadowed regions in the light plane. The light plane receivermodule's left and right receiver (Rcvr) components (i.e., opticaldetectors) convert the amount of light on the left and right sides ofthe laser split line into separate left and right electrical signals,respectively, which are digitized by receiver electronics when thereceiver electronics receives a trigger or sampling signal from encoderelectronics as described in this application.

The PC analyzes the digitized signals from the receiver electronics andcomputes left and right sensor heights. The measurement computation usesa sensor height calibration (described later) to convert the rawdigitized sensor signals into calibrated heights, measured inmillimeters.

The surface height measurement is further illustrated in FIG. 9. Theleft and right sensor heights are the amount of light blockage in thelight plane, on both left and right sides. The light plane is nominally,but not exactly, perpendicular to both the stage axis 28 and the partaxis.

Naming of Signals

In what follows the measurements that are based on light detections inthe left or right receivers are referred to as left or right sensorsignals or measurements, depending on the context. With four laser beamlines there are eight sensor signals in the system 10. When referring tothese signals in the entire system 10 the laser number is added tospecify the beam line. Thus names for the sensor signals range fromlaser-1, left sensor through laser-4, right sensor.

As previously mentioned, FIG. 8 is a block diagram which illustrates thebasic beam line system components including the laser 64 which generatesa laser beam, the mirror 62 which reflects the laser beam and the lightplane generator module 66 which generates a light plane which isprojected at the part or holding device (UUT). The part or holdingdevice blocks a portion of the light plane. The components also includethe light plane receiver module 68 having left and right receivers orphoto detectors. However, it is to be understood that instead of twodetectors, a line scan camera, an LCD camera or other optical detectordevice may be provided.

FIG. 9 which is a schematic perspective view of a cylindrical part orholding device which is intersected by a projected laser light planeillustrates light plane measurement geometry.

Stage Position of the Optical Head

The optical head 12 generates four spaced apart laser light planes andis translated up and down by the moving stage system illustrated in FIG.10. The light planes are projected perpendicular to the Z-axis definedby the movement of the stage (i.e., the stage axis 28). The linearencoder 30 preferably produces one measurement trigger signal pulseevery 4 μm of stage movement. The signal is sent to the receiverelectronics module. At the receiver electronics module, each measurementtrigger signal pulse causes each of the eight sensor signals to bedigitized and stored into the PC's memory.

This combination of the linear encoder electronics, the measurementtrigger signal, and the receiver electronics creates a sequential recordof sensor digitized raw signals when the stage moves from bottom to top.The resulting record can be interpreted as a record of light blockage byeither a part, a holding device or the calibration device at a knownseries of positions, spaced 4 μm apart. In a preferred system the totallinear length of stage movement is about 235 mm.

The position measurement system described above measures positionintervals, but generally not repeatable positions. The moving stage isstopped near the top and bottom travel limits by electronic limitswitches (not shown). Reaching the top or bottom travel limit switch bythe stage causes the motor 20 to stop. However the actual stoppedposition is only approximate, since the limit switches are not precisioninstruments calibrated to the encoder 30 and since the distance thestage requires for stopping depends on the speed of travel. In practicethis results in an uncertainty in the stopping position that can be aslarge as 500 μm.

To make a predictable starting position a light blockage signal isanalyzed to extract the index position of the beginning of thecalibration cone 40 in each sensor's digitized raw signal. The beginningof the cone 40 is preferably formed by a 3.175 mm (i.e., 0.125″)diameter cylinder 96 of the cone 40 about 6.0325 mm (i.e., 0.2375″)long. The raw sensor signal is at a high level as each light plane movestowards the beginning edge of the cone 40, followed by a sharp stepdecrease in the response, and finally followed by a constant responsealong the length of the cylinder 96. The analysis software locates themidpoint of the sharp step in the response and use that index positionto set the position zero for the sensor stage axis position. An exampleof the raw sensor signal is shown in FIG. 11.

It is found in practice that this technique can reduce the positionaluncertainty of fixed positions on the part, a holding device or on thecone 40 to an amount (5 μm) that is much less than the uncertainty inthe stopping position (300 μm).

FIG. 10 is a schematic view, partially broken away, of various positionmeasurement system components together with the optical head 12 which ismounted on the stage to move with it to scan the calibration cone 40(and later the part and the holding devices). As previously mentioned,the motor 20 is coupled to the lead screw 22 which drives the movingstage and, consequently, the optical head 12 in a direction along thestage axis 28 dependent on the direction of rotary motion of the leadscrew 22. The linear encoder senses the linear position of the movingstage along the stage axis 28 and provides a corresponding output signalto the linear encoder electronics. The electronics generate a triggersignal about every 4 μm of stage movement along the stage axis. Thetrigger signal is received and processed by the receiver electronics aspreviously described.

FIG. 11 illustrates graphs of various raw sensor signals generated atthe beginning of the calibration cone 40 (i.e., the cylinder 96). Shownare the signals from sensors laser-1, left, laser-1, right, laser-4,left, and laser-4, right. The raw sensor signals are plotted with thehighest values at the bottom of the graphs with values decreasingupwards. The left most step in the laser-1, left response represents ajump of about 2650 digitization units in the raw sensor response.

The plane of laser light from laser-4 is blocked by the calibration cone40 before the plane of laser light from laser-1 on this upward movingscan. That is because the light plane for the laser-1 is the lowest inthe optical head 12 as shown in FIG. 10.

Multiple Beam Optical Head

As previously mentioned, the optical head 12 contains four beam linesubsystems. The subsystems are aligned on a common central axis. Lookingdirectly down on the optical head 12 the beam line light plane splitlines preferably intersect at a common point as shown in FIG. 12. Theangles of the beam lines, relative to the front of the optical head baseplate, are 22.5, 67.5, 112.5, and 157.5 degrees.

This arrangement, combined with the mechanical scanning of the lightplanes, results in eight outline images of the part and the holdingdevices, one per sensor. FIG. 13 shows the geometry of a typicalsituation when light from the four beam lines (light planes) intersectsa part or a holding device with a circular cross section. For each beamline, two shadow rays graze (are tangent to) the surface of the part orholding device, marking the left and right limits of the part's orholding device's shadow. The points of intersection between the shadowrays and the part's or holding device's surface are called shadowpoints.

The calibrated distance between the shadow points and the light planesplit line is the sensor height as shown in FIG. 9. FIG. 14 shows thesensor height plotted for both the left and right sensors of laser-1 fora full scan of a threaded part. This plot is essentially an orthographicprojection of the part, in a viewing direction aligned with the laser-1,laser split line vector.

As previously mentioned, FIG. 12 is a top plan schematic view of themodules 66 and 68 of the optical head 12 with its top plate 60 removed.The laser split line for each transmitter module 66 is indicated as adashed ray which has an arrow head which illustrates the direction oftravel of the light beam and the plane of laser light.

FIG. 13 is a top plan view of a cylindrical part or a holding devicewhich is scanned by four planes of laser light including shadow rayswhich are tangent to the part at shadow points.

FIG. 14 is a graph of sensor height data created from a threaded partsuch as a standard thread plug go gage. Shown is laser-1 data; the leftsensor data is plotted on the top half of the image, the right sensordata is plotted on the bottom half of the image.

Calibration Cone

The calibration cone 40 is a device which has a precisely manufacturedshape or outer surface which is scanned to obtain calibration datawhich, in turn, is used to convert sensor raw digitized signals tocalibrated sensor height measurements. The cone 40 is a rotationallysymmetric with several distinct regions, each designed to perform adifferent calibration function.

A symbolic sketch of the calibration cone's form is shown in FIG. 15.The calibration cone is designed with usage regions, or specific shapesdesigned to accomplish specific calibration goals. These regions arelisted in Table 1 below. Each “usage region” is designed to allow aspecific piece of calibration information to be extracted from thescanned data set. The following sections briefly describe these piecesof calibration information.

Stage Position Alignment of Different Sensors

Each of the lasers 64 is mounted in the optical head 12 at a differentheight offset to the base plate 61 of the optical head 12 as describedwith reference to FIG. 10. The height offsets are not as precise asdesired and depend on detailed optical and mechanical adjustments of theoptical head 12 and its optical modules 66 and 68. The height differencebetween adjacent channels might be as large as 500 μm. The only way tomake sure that the calibrated sensor stage positions refer to the samephysical objects at the same stage positions is to measure a commonphysical position.

The precise location of the middle of the “begin cone edge” marks thecommon zero (0) of each laser sensor's calibrated stage position.

Sensor Height Zero Position Alignment of Different Sensors

It is important to select a common zero position in a plane that isperpendicular to the stage axis 28 and aligned parallel to the lightplanes in the optical head 12. The position zero that is selected in thecalibration process is the intersection of the light planes with thecenter of the small cylinder 96 at the beginning of the calibration cone40 (i.e., FIG. 10).

A light plane split line defines a natural zero for the sensor heightmeasurement, but the four light plane split lines do not necessarilyintersect in a single point as illustrated in FIG. 12. The optical headalignment process only ensures that the light plane split linesintersect within a 1/16″ (1587.5 μm) cylinder.

Adding the position offset described above makes the lines defining theposition zero of each sensor's calibrated height intersect at the centerof the small cylinder 96 at the beginning of the calibration cone 40.Measuring the center, after calibration, typically gives a centrallocation that is less than 1 μm from (0,0).

Calibration Cone Aspect Vector, Relative to Stage Axis

The calibration cone's central axis is not necessarily exactly alignedwith the axis 28 of the stage motion, due to tolerance stackup on a longpath from cone 40, to part holder, to base plate, to slide support,etc., and finally to the slide.

By measuring the sensor heights of the center of two calibration coneregions, “const diam-1” and “const diam-2”, the inclination of thecalibration cone's aspect vector relative to the stage axis 28 can bedetermined.

Typical measured angles for the calibration cone aspect vector relativeto the stage axis 28 are in the range (0 . . . 1 degree) or (0 . . .17.5 mrad).

Light Plane Angle, Relative to Calibration Cone

The laser light planes are not exactly perpendicular to the calibrationcone aspect vector. In order to know the angle between the light planesand the calibration cone 40, the “multi-step” region is analyzed. Signalprocessing software can very accurately measure the position of each oneof the set of 5 step edges in the “multi-step” region. The distancebetween the 5 step edges is precisely known. With this information theangle of the light plane relative to the cone aspect vector can becomputed. This angle is important in determining exactly how the lightplane intersects the calibration cone 40 and thus in extractingcalibration information from the data.

Typical measured angles of the light plane relative to the calibrationcone 40 are in the range (0 . . . 0.75 degree) or (0 . . . 13.1 mrad).

Sensor Height Calibration

The output of the laser scanned measurement is a record of the sensordigitized raw signals for each sensor. To make sensor heightmeasurements in physical coordinates the raw signals need to beconverted to sensor heights.

Two regions on the calibration cone, “const slope-1” and “const slope-2”provide this information. For example, the diameter of the intersectionbetween a laser light plane and “const slope-1” region varies between0.125″ and 0.750″. The exact diameter can be computed by knowing thedistance between the laser light plane and the beginning of the “constslope-1” region, since the region is manufactured to high precision.

Based on the diameter of intersection and the laser sensor outputs,calibration tables may be constructed to convert digitized raw signalsto calibrated sensor heights.

Measurement of the Sensor “No Blockage” Signal Level

The raw signal level with no cone 40 or UUT in the light plane sensorbeam is also measured. For small parts or ammunition cases it is oftenrequired to extrapolate the sensor raw signal to sensor heightconversion table to smaller heights than are measured on the cone. Theextrapolation is carried out with less accuracy than more directmeasurements, but the extrapolation is very useful, especially for partsthat are only slightly smaller than the begin cone cylinder 96 (“constdiam-0” region) of the cone 40 or parts that are offset from thecalibration cone central axis.

The “no blockage” signal level is also required in order to correctlyfind the beginning of the cone 40.

Finally, excessive variability of the “no blockage” signal level is asignature of variability of the light output of the laser 64 in a beamline. This variability of the “no blockage” signal level is monitored togenerate a signal which indicates that the apparatus which generates thebeam line requires repair or is temporarily unable to carry out highprecision diameter measurements.

This measurement goal does not involve measurement of the calibrationcone 40. However, it is required to interpret calibration cone analysis.It is made possible by the physical design of the part holder base asillustrated in FIG. 3 a, and by the alignment of the bottom position ofthe optical head 12 on the base/slide to the part holder base.

TABLE 1 Calibration Cone - Usage Regions (i.e., FIGS. 15 and 26) RegionDescription begin cone edge Begin cone step edge. Measurement of theprecise step edge position makes it possible to align measurements madewith different laser sensors. The precise edge location is each sensor'sstage axis zero (0) coordinate. const diam-0 Constant diameter region-0,diameter 0.125″. The central axis of this region, projected to thesensor is the common physical sensor height zero (0) coordinate. constdiam-1 Constant diameter region-1, diameter 0.750″. The central axis ofthis region is utilized for estimates of the 3-D cone aspect vector,relative to the stage axis 28. const diam-2 Constant diameter region-2,diameter 0.750″. The central axis of this region is utilized forestimates of the 3-D cone aspect vector, relative to the stage axis 28.const max diam Constant maximum diameter region, diameter 1.500″. Marksthe boundary between the constant slope region and the multiple stepregion. const slope-1 Constant slope region-1. Measurements in thisregion establish the raw sensor output corresponding to diameters in theinterval 0.125″ and 0.750″. const slope-2 Constant slope region-2.Measurements in this region establish the raw sensor outputcorresponding to diameters in the interval 0.750″ to 1.500″. multi stepMultiple step region. Measurements of the precise step edge positions ofthe five equal height edges allow a rough estimate to be made of theangle of the Laser Light Plane relative to the cone aspect vector. slopeedge-0, 1 Slope edges 1 and 2. The edge positions mark the boundaries ofthe const slope-1 region. slope edge-2, 3 Slope edges 2 and 3. The edgepositions mark the boundaries of the const slope-2 region.

Other embodiments of the calibration cone 40 are possible while stillmeeting the general measurement goals of the Laser Lab system 10.

Calibration Measurement Goals

These are goals that calibration analysis of the cone data meet:

1) stage position alignment of different sensors

2) sensor height zero position alignment of different sensors

3) calibration cone aspect vector, relative to stage axis 28

4) light plane angle, relative to the calibration cone 40

5) sensor height calibration

6) measurement of the “no blockage” signal level.

In another embodiment of the calibration cone 40, changes to overallminimum and maximum width of the cone 40 can be made.

The calibration cone's “point design” allows extraction of calibrationdata relevant to measuring parts and holding devices in the diameterrange 0.125″ to 1.500″.

The system can be designed for smaller or larger parts wherein widthmeasurement limits could be changed. A small compact system formeasuring a range of smaller diameters could utilize a calibration cone40 with minimum and maximum diameters of 0.065″ and 0.500″ for example.

In another embodiment, changes to overall length of the cone 40 can bemade.

The calibration cone's “point design” is specified as a compromisebetween ease of analysis and physical compactness. The lower the slopein the “const slope” regions, the more precise the data that isextracted. This is due to two reasons. First, dividing up a slopingregion into “bins” and then determining the raw data to heightconversion factor within an individual bin is more accurate when themechanical diameter varies least within the bin.

Second, inaccuracies in determining the light plane twist angle or lightplanes that are not flat are multiplied into inaccuracies in the rawdata to height conversion factors by the mechanical slope of the constslope regions in the calibration cone 40.

In yet another embodiment, changes to slope in “const slope” regions canbe made. As noted in the above-noted discussion, lower slopes in the“const slope” regions translates to more accuracy in the sensor rawdata-to-height conversion factor tables.

A “point design” directed towards the goal of smaller parts orammunition cases and towards a more compact system design might have acone 40 with the same length, but 3× smaller width dimensions. Thiswould allow more precision in measuring smaller parts or ammunitioncases with smaller width light planes.

In another embodiment, the cone 40 can be supported either “point down”or “point up”. The mounting direction does not matter, the calibrationmeasurement goals can be met with either orientation. However, themounting method should still allow a region of “no blockage” sensorsignal to be measured during each up/down scan.

In yet another embodiment, different number of steps in “multi-step”region can be provided. The number of steps in the “multi-step” regioncan be varied and the dimensions of the steps can be changed.

The calibration analysis that determines the light plane twist angleuses the difference in position between steps detected in the left andright sensors. Having more steps makes the determination more precise.

In another embodiment, changes to the number, position, or diameter ofthe “constant diameter” regions can be made. The calibration cone aspectvector is measured by analysis of the central axis of two constantdiameter regions, “const diam-1” and “const diam-2”. Each region is thesame diameter. Determination of the location of their 3-D center allowdetermination of the central axis of the calibration cone 40, from thetwo 3-D center points. It is important that both regions be the samediameter, to minimize the effect of diameter measurement errors on thecalibration cone axis vector.

The regions could be a different diameter, either smaller or larger.There could also be more than two regions. Then a line could be fitthrough three or more 3-D points to determine the calibration cone axisvector. It is important that there be at least two regions, one regiondoes not typically determine the calibration cone axis vector.

Laser Lab Calibration Analysis Overview

What is now described is a process oriented overview of the Laser Labcalibration procedure. In this procedure raw sensor data and geometricdescriptions of the calibration cone 40 are utilized to producecalibration data. This data set can be used later to produce calibratedsensor data from raw sensor data, as described in the “data processing”section, below.

Calibration Goals

As previously mentioned, the following is a partial list of themeasurement process goals that are met by the design of the mechanicalcalibration cone 40 and also by the calibration data analysis procedure:

G-1: stage position alignment of different sensorsG-2: sensor height zero position alignment of different sensorsG-3: calibration cone aspect vector, relative to stage axis 28G-4: light plane angle, relative to calibration cone 40G-5: sensor height calibration

Data Processing

After a laser lab scan of a UUT, eight sensor digitized raw signals arestored, one each for the left and right sensors, repeated for each offour light planes. Each of these digitized raw sensor signals is asingle array or indexed list. The sensor raw signal array has an indexfor every sample stored on the laser lab scan of the UUT; index 1999refers to the 1999-th sample taken during the scan. The value in thearray at index 1999 is the value of the 1999-th raw signal sample.

Production of Calibration Data

After the raw sensor signal arrays are stored in memory of the computeror PC, the arrays are analyzed to extract calibration information. Thisinformation is extracted from the part of the digitized raw sensorsignals that contains the image of the calibration cone 40. Theextracted information results in a number of tables and parameters,collectively called “calibration data”.

Production of Calibrated Sensor Data

Once calibration data has been successfully calculated it is utilized toproduce a new set of vectors called “calibrated sensor data”. The newcalibrated sensor data vectors contain two pieces of information at each“index”: the pair (calibrated stage position, calibrated height).

Calibrated Stage Position

Stage position for the N-th index in the calibrated sensor data vectoris the distance (in millimeters) from the beginning of the calibrationcone 40 to the position where the raw sensor signal at the N-th indexwas taken.

First the raw sensor stage indices (the index of the raw sensor signalvector) are multiplied by the stage linear encoder spacing (4 μm in thecurrent system), producing raw sensor stage positions. Then the rawstage positions are referenced to the position of the beginning of thecalibration cone 40.

Finally the raw stage positions are corrected for laser tilt. The tiltcorrection depends on the height of the sensor data point. If the laserplane is slightly tilted, then any non-zero sensor height alsorepresents a slight change of stage position since the light plane,stage axis coordinate system is not orthogonal. After the correction thecalibrated sensor height, calibrated sensor stage position coordinatesystem is orthogonal. Having an orthogonal coordinate system makes latermeasurement analysis much simpler.

Calibrated Sensor Height

Calibrated sensor height for the N-th index in the calibrated sensordata array is the distance (in millimeters) from the center of thecalibration cone's beginning cylinder 96 to the shadow ray that producedthe raw sensor signal at the N-th index.

As discussed above, the observations in the non-orthogonal stageposition, light plane coordinate system are corrected for the effects oflaser tilt.

Decimation to a Uniform Sampling Interval

The corrections for laser tilt result in an array of calibrated sensordata where the stage position distances between adjacent index positionsin the vector can vary around an average value of 4 μm.

Since uniformly sampled data is much easier to work with for measurementanalysis, the calibrated sensor data vector is decimated or sampled touniformly sampled calibrated sensor data vector for measurementprocessing. In the current system the data is decimated to the originallinear encoder sample spacing of 4 μm.

FIG. 16 is a schematic block diagram which illustrates sensor dataprocessing. The figure shows the processing of raw sensor data withcalibration data to obtain uniformly sampled calibrated sensor data.

Calibration Analysis

Calibration analysis refers to the analysis of the raw sensor datavectors containing an image of the calibration cone 40. The output ofthe analysis is a set of tables and parameters called calibration data.

Rough Edge Processing

Rough edge processing discovers the presence and rough parameterizationof the signal edges in the raw sensor data.

Rough edge processing attempts to find the pattern of edges thatidentify the calibration cone 40 in the raw sensor data vector. Thispattern is schematically illustrated in FIG. 15.

Two types of edges are found. The first type of edge, a “step edge”,represents a vertical segment on the calibration cone 40. A step edgedetector finds one edge corresponding to the begin cone edge and fiveedges corresponding to the locations of the vertical segments in thecalibration cone's multi-step region.

A second type of edge, a “slope edge” represents the location where twostraight segments join with each segment having a different inclinationto the vertical. A slope edge detector looks for slope edges only inlocations where there is not a step edge. The slope edge detector findsa first unique slope edge at the location where “const diam-0” regionmeets “const slope-1” region, and in three other places.

Rough Edge Processing—Outputs

-   -   rough locations for step edges.    -   rough locations for slope edges.    -   confirmation that the calibration cone edge pattern is present        in the data.

When the rough edge processing step does not find the calibration coneedge pattern, the calibration analysis process is stopped.

Precise Edge Processing

Precise edge processing finds the exact locations of step edges in thecalibration cone edge pattern. Precise edge processing uses outputs fromrough edge processing to determine initial estimates for the edgelocations, which it then refines.

A detailed description of precise edge processing is located in AppendixB.

Precise Edge Processing—Outputs

-   -   precise location for begin cone step edge.    -   precise locations for five edges in “multi step” region.

The knowledge of the set of eight begin cone step edges, one for eachsensor, completes calibration goal G-1: stage position alignment ofdifferent sensors. This data is stored in the calibration data and usedto convert raw sensor data to calibrated sensor data.

Data Binning

At 4 μm per sampled point, there can be too much data to be effectivelyanalyzed for certain calibration processes. Data binning is the processof dividing up a set of sampled points, grouping the set into a smallerset of “bins”, each bin containing a number of adjacent sampled points.

For the tables relating the raw sensor data to calibrated sensor heightsbinning is utilized. For example, the “const diam-1” region on thecalibration cone 40 is about 12 mm long, ranging from 3.810 mm to 15.558mm along the cone axis. This would be about 3000 data points withoutbinning. At the nominal bin size of 0.2 mm this works out to about 60bins.

Another advantage of binning is that the data within the bin can beaveraged and checked for consistency.

Finally, the data bins are not constructed within a “guard” regionwithin 0.2 mm. of a detected edge. For the “const diam-1” region theedges “slope edge-1” and “slope edge-2” mark the boundaries of theregion and the “guard” region assures us that the boundary data bincontains only data from the uniformly sloping region.

Data Binning—Outputs

Four sets of data bins are produced, for each of the eight sensors:

-   -   two sets of data bins for the regions “const slope-1” and “const        slope-2”.    -   two sets of data bins for the regions “const diam-1” and “const        diam-2”.

The “const slope-n” data is used in the construction of the sensorheight calibration table. The “const diam-n” data is used as input datato the process that finds the position of the calibration cone's 0.750″diameter cylinder, for the cone aspect angle estimation process.

Laser Roll

Laser roll processing finds the angle between a light plane and thecalibration cone 40 in each laser's calibrated sensor coordinate system.

For each laser the precise edge locations for the five edges in the“multi step” region are obtained, one set for the left sensor andanother set for the right sensor. The difference between the left sensorand the right sensor edge positions can be used as input to a leastsquares estimate of the laser roll angle.

The detailed method of estimation is described in Appendix B.

The estimate of the laser roll angle assumes that the light plane isflat.

Laser Roll Processing—Outputs

-   -   four laser roll angles, relative to calibration cone 40,        projected to each laser's coordinate system.

The estimate of four laser roll angles completes calibration goal G-4:light plane angle, relative to calibration cone 40. These angles arestored into the calibration data.

Sensor Blockage and Tilt

The primary goal of the sensor blockage and tilt process is to generatea calibration table that relates the sensor raw signal values to thecalibrated sensor heights.

Achieving the primary goal is made difficult because the calibrationcone 40 may be mounted at an angle that may not be parallel to the stageaxis 28. If the cone angle is not parallel to the stage axis 28 then theinterpretation of exactly where the light plane hits the calibrationcone 40 depends on the angle between the calibration cone 40 and thestage axis 28.

To solve this problem an iterative process was created.

First, the sensor calibration tables were created, assuming the coneangle and stage axes 28 are parallel. Then using the newly createdsensor calibration table an estimate of the cone angle was made. Theprocess is repeated four times. The iterative process has been found toconverge in all cases. It is recommended that the mechanical alignmentof the cone aspect angle to the stage axis 28 be less than (1 degree).

This process is documented in more detail in Appendix B.

Sensor Blockage and Tilt—Outputs

The sensor blockage and tilt calibration process has two outputs.

-   -   eight sensor raw data to calibrated sensor height tables, stored        in a calibration object.    -   a 3-D estimate of the angle between the calibration cone aspect        angle and the stage axis 28.

The eight calibrated sensor height tables complete calibration goal G-5:sensor height calibration. The 3-D cone aspect angle meets calibrationgoal G-3: calibration cone aspect vector, relative to stage axis 28.

Sensor Height Table Extrapolation

The tables that correlates raw sensor data to calibrated sensor heightsmay need to be extended. Sometimes a small part or ammunition case witha center offset has a sensor height that is smaller than the minimumheight in the table. There are also gaps in the data, due to thepresence of “guard” regions, as discussed in this application.

Data gaps are addressed by a linear interpolation method.

For sensor heights that are smaller than the minimum sensor height inthe sensor height calibration table the table is extrapolated to a zeroheight. The last 10 points in the sensor height calibration table arefit to a line. Then additional points are added to the sensor heightcalibration table between the table's minimum height and a height ofzero.

The same process is carried out to extrapolate the sensor heightcalibration table to the maximum sensor height allowed (19.05 mm).

Sensor Height Table Extrapolation—Outputs

The outputs of the process are additional sensor height calibrationtable entries, generated from linear extrapolations to zero height andto maximum height.

Sensor Height Table—Zero Height Position

For each sensor calibration table an offset is computed to ensure thatthe sensor height is zero when the sensor views the calibration cone'sbegin cone cylinder 96 (const diam-0 region).

Sensor Height Table—Zero Height Position—Outputs

-   -   sensor height zero offset for eight sensors

This data meets calibration goal G-2: sensor height zero positionalignment of different sensors.

Thread Signal/Data Processing Introduction

What follows is a description of the structure of the thread parameterestimation process. This process provides one embodiment of the standardthread measurement “feature” in the system 10.

Thread Signal Processing

Thread signal processing is the process of estimating the followingthread parameters:

1) pitch2) major diameter3) minor diameter4) functional diameter5) lead deviation6) pitch diameter.

The input data to the process is “calibrated part data”. This data setconsists of eight vectors, one for each photodetector or sensor. Eachvector consists of an indexed table of elements, each containing a (z,h)pair. Each (z,h) pair measures the position of the UUT's shadow ray in acoordinate system that represents each sensor's view of the UUT. z is ameasurement of calibrated sensor stage axis position, and represents thedistance along the stage axis between the current data point and thestage position where the light plane hits the beginning of thecalibration cone. h is a measurement of calibrated sensor height andrepresents the distance between the middle of the beginning cylinder ofthe calibration cone and the shadow ray, perpendicular to the stageaxis.

As the thread signal processing proceeds, a number of intermediate dataproducts are produced in early processing stages that are furtheranalyzed in later stages. These include:

-   -   rough pos/neg crossing locations    -   rough crest locations    -   wire position search intervals    -   left/right flank lines    -   wire positions    -   precise crest/root locations    -   3-crest average/median measurements of major diameter, minor        diameter, pitch diameter    -   3-D crest cylinder axis    -   wire position projections on the 3-D crest cylinder axis    -   3-D crest cylinder diameter    -   3-D crest root-mean-square distance between crest data and fit.

These intermediate data products are analyzed to produce final estimatesof the thread parameters. For example major diameter is estimated astwice the radius of the 3-D crest cylinder. The 3-D crest cylinder axisthen depends on the precise crest/root locations. The crest/rootlocations then depend on the search intervals based on rough crestlocations and pos/neg crossings, and on data from the originalcalibrated part data.

Processing Restrictions

Inspection Region

The thread processing occurs between stage position limits called aninspection region. In the Laser Lab template editor, the user specifiesthe inspection region by manipulating the upper and lower stage positionlimits, overlaid on an image of the part.

These limits use the calibrated sensor stage position so thatmeasurements by different lasers are aligned to the approximatelysimilar physical positions on the part.

The estimation of thread parameters is specified to be an averageestimate over all the data within the inspection region. In practice,some of the intermediate data products are estimated outside of theinspection region in order to allow estimation of all thread parameterswithin the full region. For example, a wire position within theinspection region may require a thread crest outside the inspectionregion.

Measurement Assumption for the Inspection Region

The following requirements guide the user's placement of the inspectionregion on the image of the part. At present the analysis software doesnot detect a failure of any of the listed requirements directly.

The first assumption is that the thread parameters be constantthroughout the inspection region. This enables the software to averagethe estimates from different positions within the inspection region andnot be concerned with partitioning or segmenting the data into differentregions for special processing.

This requirement excludes the following types of data from theinspection region:

-   -   the beginning or end of a threaded region, with thread crests        less than full height.    -   a threaded region with a taper.    -   a threaded region with a notch or extensive damage.

A second assumption is that the inspection region contains at least fourthread pitches. This amount of data is required to construct several ofthe intermediate data products with the required accuracy. Theintermediate data product that is most closely tied to this requirementis the 3-D peak cylinder described in this application.

A third assumption is that the thread be manufactured with a 60-degreeflank angle. Thread processing implicitly uses this parameter in severalplaces. One of the most direct usages is the conversion of leaddeviation into functional diameter. Other flank angles or other threadform shapes would require different procedures.

A fourth assumption is that the thread has a cylindrical cross section.Non-cylindrical threads would require the 3-D peak cylinder to besuitably generalized. Incorrect fit to a non cylindrical cross sectionwould lead to incorrect lead deviation measures in the currentimplementation.

A fifth assumption is that the thread has a single helix. Currentlydouble threads are not supported.

The software does not check the assumptions. Failure to meet therequirements will typically lead to bias in the thread measurement, orin a failure to successfully measure the inspection region.

In practice, these requirements exclude the measurement of the followingobjects:

-   -   non-standard thread types, especially self-tapping screws.    -   small threaded regions with 2 or 3 pitches.    -   Taptite trilobe threaded regions.

Rough Crossings

The thread model described below is a sampled representation of onesensor's thread profile, for exactly one pitch. The thread model startsat the midpoint of a rising thread flank and ends one pitch later.

Using a correlation detector the thread model is matched to data withinthe inspection regions, producing thresholded detections within theinspection region, that are called crossings. FIG. 17 shows a sketch ofa thread model matched to the sensor data.

Later processing refinements noted in this application may make thecrossings more accurate. The refinements also separate the crossingsinto positive crossings (right flank line in FIG. 17) and negativecrossings (left flank line in FIG. 17). FIG. 18 illustrates selectedconcepts of a thread form. The thread model is a lateral sequence ofpoints that represent a best estimate of the outline of one cycle of thethread form.

Rough Crest and Root Positions

A crest/root detector extracts rough crest and root positions betweenthe matched adjacent pairs of positive and negative crossings.

Pitch Estimate

A pitch estimate is required for step set gage wire diameter. Theestimate is required to be accurate enough to unambiguously select aunique gage wire from the set appropriate for the measurement. Thecurrent process uses a two-stage process.

This process may be simplified as described in this application.

First Estimate

Crossing data is analyzed and averaged over all sensors to create athread pitch estimate, the “crossing pitch”.

Second Pitch Estimate

The steps: set wire gage diameter, wire position search intervals,measure flank lines and measure 3-point diameters noted below arecompleted in a first iteration. Then the wire positions are averagedover all sensors and positions to compute a pitch estimate.

Set Gage Wire Diameter

Gage wires are utilized in physical thread measurements of pitchdiameter in the prior art. Two wires are placed in adjacent threads onone side of the UUT, and a single wire is placed on the other side ofthe UUT. A micrometer measures the distance between the reference lineestablished by the two adjacent gage wires and the reference pointestablished by the other gage wire. A tabulated correction formulaconverts the micrometer distance to an estimate of the pitch diameter.

Gage wire sizes are thus selected prior to the thread measurement. To dothis, the thread pitch is estimated as previously described and then theclosest gage wire in a set is selected to the pitch estimate. The gagewire set that is used is the one appropriate to the type of measurement;currently there is one set for the metric coarse thread sequence, andanother for a similar English thread set. The gage wire sets are chosenat part template edit time.

Wire Position Search Intervals

Virtual gage wires are placed onto the calibrated sensor data throughoutthe inspection region. In order to place the virtual gage wires we mustidentify search intervals for each wire to be located.

A requirement of the following processing steps is that the wirepositions in the inspection region have no gaps. Another requirement isthat a wire position search interval consist of two valid thread crests,one valid thread root between the two thread crests, and validpositive/negative crossings between the crest/root pairs.

The set of positive/negative crossings and crest/root positions aresearched for the set of wire position search intervals to analyze. Theresult is a set of intervals, one set per sensor.

Measure Flank Lines

FIG. 19 shows a sketch of a portion of calibrated sensor data in asingle wire position search interval.

The specification of a valid wire position search interval means thatthe form of the calibrated sensor data is approximately as shown in FIG.19. This form was used to create a plan to robustly extract flank linedata.

For the left flank line (example) we analyze all data between the roughpositions of the left crest and the central root. The height limits of aflank line data extraction region that covers 70% (a configurableparameter) of the height interval between left crest and central rootare then determined. This data is extracted into a data set and fit to aline, becoming the left flank line.

The procedure avoids the non-linear regions near the left crest andcentral root. In addition a “flank line valid” flag is computed, basedon the RMS distance between the left flank line and the data within theleft flank line data extraction region. If the RMS distance between theflank line and the data points in the flank line data extractioninterval is larger than 10 μm per point (a configurable parameter), thenthe flag is set to invalid.

The process is repeated for the right flank line and then for all wireposition search intervals.

Measure Wire Positions

The wire positions are calculated, given the left and right flank linesand the wire size. As shown in FIG. 19, the virtual wire is tangent toeach flank line and the resulting position is calculated with a simplegeometric formula.

The position has a valid flag that is true when both flank lines arevalid, and false otherwise.

Measure 3-Point Diameters

The 3-point technique is a method to measure the minor, major, and pitchdiameters without explicitly utilizing 3-D information. All computationsare carried out in the 2-D laser sensor coordinate system.

For example, consider the major diameter. It is defined as the diameterof a cylinder that contains all the inspection region's thread crests.

In this method, the top of a thread crest in calibrated sensor (stageposition, height) coordinates forms an elementary measurement. Theelementary measurements are combined into triplets for further analysis.Only crests from the two sensors of a single laser are combined.

Two adjacent thread crest positions in sensor-1 are combined with thethread crest position in sensor-2 that is closest to the averageposition of crests in the first sensor. The two crests in sensor-1 forma reference line. Then the distance from the reference line to the crestin sensor-2 is computed. This is the 3-crest distance for that cresttriplet.

In this manner, the 3-crest distances from all adjacent crest tripletsare computed, for all laser data. The 3-crest diameter measurement iseither the average or the median of all the 3-crest distances within the3-crest data vector.

3-Point Minor Diameter

The 3-point minor diameter computes 3-point distances using precise rootlocations in the sensor data. The 3-point minor diameter is the averageof the 3-point distance vector.

3-Point Major Diameter

The 3-point major diameter computes 3-crest distances using precisecrest locations in the sensor data. The 3-point major diameter is themedian of the 3-point distance vector.

3-Point Wire Pitch Diameter

The 3-point pitch diameter computes 3-point distances using the wirepositions computed in the sensor data. The 3-point wire pitch diameteris the median of the 3-point wire pitch diameter. FIG. 20 is a schematicview that illustrates a 3-point distance method, applied to thread wirepositions. Shown are two wire positions in the top thread form with areference line drawn between them. Also shown is a single wire positionon the bottom thread form with the 3-point distance indicated.

FIGS. 21 and 22 are screen shots from a user interface of a PC whichillustrate intermediate data extracted from a M16×1.5 thread plug gage.FIG. 22 is an enlarged view with its focus on a single thread pitch.

Measure 3-D Crest Cylinder

The measured thread crest position data is analyzed to obtain a 3-Dcylinder with least squares methods. A mathematical description of themethod is given in Appendix C.

The 3-D crest cylinder fit has several output parameters of interest:

-   -   the RMS distance between the crest position data and the fitted        shape.    -   the 3-D location of the cylinder's central axis.    -   the radius of the cylinder.

Project Wire Positions onto 3-D Crest Cylinder Axis

Measured wire positions can be combined with the 3-D location of the 3-Dcrest cylinder's central axis. An imaginary disk, perpendicular to thecylinder axis that goes through the measured wire position marks aposition on the 3-D crest cylinder axis.

A data set consisting of the projections of all sensor wire positions isconstructed.

For a perfect helical thread and for perfectly measured wire positionsthe spacing between the positions in the projected wire positions shouldbe exactly P/8, where P is the pitch of the thread. The eight sensorseach give a view that is rotated ⅛ revolution between adjacent sensors.

For a right handed thread, the wire positions project onto the axis atincreasing positions in the order L1L, L2L, L3L, L4L, L1R, L2R, L3R,L4R, and then L1L, . . . , etc.

The output intermediate data is a vector, sorted from minimum to maximumsensor stage position of the projected wire positions. In addition eachwire position data item is annotated with labels that specify the laserand sensor that produced the data item and other labels containingadditional information.

Thread Parameter Estimation

Thread parameter estimation uses the intermediate data products and mayalso correct them based on a model of the measurement, prior toproducing a final thread parameter estimate.

Wire Pitch

Thread pitch is estimated from the wire center intermediate data. Foreach sensor data set, the adjacent pairs of wire positions are used tocalculate an adjacent wire pitch, one per adjacent wire positions. Forall lasers, each wire pitch is added to a wire pitch vector.

The wire pitch estimate is the median of the elements in the wire pitchvector.

Major Diameter

Thread major diameter is typically reported as the diameter of the 3-Dcrest cylinder.

If the 3-D crest cylinder fit was unsuccessful, the major diameter isestimated in a different way, detailed below. The cylinder fit can faildue to several factors:

-   -   part inclined at too great an angle with respect to the stage        axis.    -   thread crest positions do not fit a cylinder, the RMS        fit-to-data distance is too large.

When the cylinder fit fails the major diameter is estimated from the3-point major diameter data. This case is special because a previouscondition (cylinder fit) has already failed. We found in practice thatthe cylinder fit most often failed when the threaded region was tooshort or the inspection extended beyond the end of the threaded region.

Because of this bias we found that a simple median of the 3-point majordiameter data would typically be too low, most of the good 3-point datawas concentrated at the highest measurements. In this case the majordiameter estimate is the value such that 20% of the 3-point data ishigher and 80% of the 3-point data is lower.

Calibration Correction

Major diameter is also corrected by a final end-to-end calibration ofthe total system. The reported major diameter is often too low, withbias ranging from −20 μm to 0.

After diameter calibration we expose the system to a set of measuredthread plug gages. Their major diameter bias is then plotted as afunction of diameter and a simple segmented line is fitted to the biasresults. These bias fits then are entered into the system configurationfile and are used to correct the measured major diameter with themeasured bias.

Minor Diameter

Thread minor diameter is estimated with the 3-point minor diameterdistance vector. The minor diameter value is the average of the elementsin the distance vector.

Pitch Diameter

Pitch diameter estimation uses two sets of intermediate data products,the wire positions and the 3-D crest cylinder fit.

The pitch diameter estimate calculation is presented in a step-by-steplist below:

a) Compute the pitch diameter contact points with the thread flanks bycalculating the intersection of the wire shape with the left or rightflank lines.

b) Average the left and right points of intersection, and compute thedistance (radius) from the average point to the 3-D crest cylinder fitaxis. This is the pitch diameter radius for each wire position.

c) Calculate the average value of the pitch diameter radius for eachsensor.

d) Correct each sensor's average wire position radius for the partprojection angle, using the angle of the 3-D crest cylinder axis to thestage axis, projected into each sensor's coordinate system.

e) Add left and right sensor corrected pitch diameter radius estimatesto produce an estimate of the pitch diameter for each laser.

f) Average the laser estimates to produce the system pitch diameterestimate.

Correction for Part Projection Angle

The computation of pitch diameter is complicated by projection effects.The laser light performs an almost perfect orthographic (shadow)projection of the thread's shape. However the projection is not the samething as the thread cross section, which is specified in thread designdocuments. The cross section is the thread shape if it were cut by aplane going through the thread's central axis.

The difference is caused by the thread lead angle, which is in the rangeof 1-3 degrees for many typical threads. The lead angle means that thethread cross section is most accurately viewed in shadow when theviewing direction coincides with the direction of the lead.

It is impossible to position the thread so that a shadow view of thethread is simultaneously aligned with the top and bottom threads. Forthe example of a thread with a 3 degree lead angle, tilting the threadto align the top of the thread with the viewing angle will make theangle between the lead and the viewing angle for the bottom thread about6 degrees.

A correction factor was developed for this effect. If the tilt of thethread with respect to the viewing angle is known, the observed pitchdiameter radius for the expected bias caused by the projection angle canbe corrected. This correction is precomputed and stored in a table.

For each sensor the tilt of the thread with respect to the viewing anglecan be obtained from the 3-D cylinder fit axis. Separate corrections areapplied to the left and right sensors.

Calibration Correction

Pitch diameter is also corrected by a final end-to-end calibration ofthe total system. The reported pitch diameter is often too high, withbias ranging from +5 μm to +35 μm.

After diameter calibration, the system is exposed to a set of measuredthread plug gages. Then their pitch diameter bias is plotted as afunction of diameter and fit a simple segmented line to the biasresults. These bias fits then are entered into the system calibrationfile and are used to correct the measured pitch diameter with themeasured bias.

Lead Deviation

The lead deviation estimate uses the wire pitch and the locations of thewire positions as projected onto the 3-D cylinder fit axis.

For an ideal helical thread, the wire position projections should resultin a regular pattern along the 3-D cylinder fit axis. The projection ofthe first laser-1, left, wire position should lie about (⅛) pitch fromthe projection of the first laser-2, left, wire position. Lead deviationis the deviation of that pattern from the ideal, measured as a maximumdistance of any projected wire position from the ideal pattern.

The computation of the lead deviation estimate follows a step-by-stepprocedure:

a) Create a wire position projection vector, containing all the data.

b) Sort the wire position projection vector in order of position alongthe 3-D cylinder fit axis.

c) Convert the wire positions of the elements of the vector intodegrees, by multiplying by the factor (360/pitch) and then reducing theelement values modulo 360.

d) Calculate an offset value so that the maximum absolute value of thedegree-valued element positions is minimal. For example with a leaddeviation of 0.010 mm for a 1 mm pitch thread, the absolute value of atleast one degree-value element position would be 3.60 degrees. (0.010mm/1 mm equals ( 1/100) and 360/100 is 3.60.)

e) Convert the value from degrees to millimeters and report as the leaddeviation estimate.

Note that all lead deviation estimates are positive.

Calibration Correction

Errors in measurement mean that the physical measurement of a perfectthread will have a positive lead deviation.

To attempt to correct for this effect, the lead deviation for a set ofthread plug gages is measured and plotted as a function of gagediameter. The most common form observed is a constant lead deviation of0.010 mm. to 0.020 mm.

This value observed in calibration with thread gages is taken to be abias. This amount of bias is entered into the system calibration fileand used to correct the measured lead deviation for this measurementbias.

Functional Diameter

Functional diameter is currently defined in practice by the fit of aspecial fit gage over the thread. The special fit gage is essentially anut that is split in two by a plane cut through the central axis of thenut. The two halves of the fit gage are held in a fixture that measuresthe distance between the two halves. There is one special fit gage forevery thread type.

Functional diameter is defined as the pitch diameter when the specialfit gage is clamped tightly over a thread plug setting gage. When adifferent UUT is put into the fit gage the fit gage may expand slightly,due to a summation of effects involving the differences between the UUTand the thread plug setting gage used to setup the functional diametermeasurement. The functional diameter measurement is then the thread plugsetting gage's pitch diameter plus the additional separation between thetwo fit gage pieces.

Functional Diameter—Laser Lab Estimator

In the Laser Lab, our functional diameter measurement method is anapproximation of the fit gage method. A full 3-D analog of the physicalfit gage is not performed. Instead, an approximation is made thatinvolves the use of lead deviation and the shape of the thread form.

If we imagine the thread form as perfect and also having a 60 degreeflank angle then lead deviations should cause a the thread form fit gagepieces to move apart. A single lead deviation either up or down thethread form axis will cause a single split piece of the fitting gage tomove outward. The amount of outward movement for a 60 degree flank anglewill be equal to (√{square root over (3)}) (lead deviation). Themovement provides a clearance for both positive and negative movementsof the lead, relative to a perfect helical shape.

The Laser Lab estimator for functional diameter, (FD) is given below:

FD→PD+√{square root over (3)}(LeadDeviation).

Learning the Thread Model

The thread model is a learned sequence of points that represent a bestestimate of the outline of one cycle of the thread form. The threadmodel is calculated when the inspection region is specified, at templateedit time.

The measure template routine uses a pattern match algorithm with a sinewave pattern to identify periodicity in the inspection region data. Thisprocess determines an approximate thread pitch. The process alsocalculates a starting point in the data vector for the first beginningof the matched pattern, which is an approximation to the first midpointof a right flank line.

With the pitch and the starting point in hand, the measure templateroutine can then calculate an average thread model. Starting with thefirst sample point in the matched pattern, points that are 1,2,3, . . ., N pitches later in the inspection region are averaged to form thefirst point of the thread model. The process is repeated for all therest of the points in the first matched pattern. The thread model isthen stored in the template for later use.

The following is a description of the structure of the trilobe ortrilobular estimation process.

Trilobe Signal Processing

Trilobe signal processing analyzes calibrated part data within theinspection region and produces intermediate data products that areanalyzed by the trilobe parameter estimation process described below.Eight values are produced in trilobe signal processing, four laser-ndiameters and four laser-n centers.

Intermediate Data Product Description Laser-n diameter Distance fromleft shadow ray to right shadow ray Laser-n center Midpoint of intervalspanned by left and right shadow rays.

Trilobe Blank Signal Processing

For the trilobe blank, the laser diameter and center are estimated assimple averages of calibrated sensor data within the inspection region.

The laser-n diameter is the average of the mean left sensor height andthe mean right sensor height.

The laser-n center is the difference of the mean right sensor height andthe mean left sensor height.

Trilobe Threaded Region Signal Processing

For the trilobe threaded region, the parameters of a trilobe cylinderare estimated that touches all the thread crests within the threadedregion.

This process can be subdivided into three parts:

-   -   obtain thread crest locations from thread region global object    -   estimate sensor height from sensor thread crests, robustly    -   compute laser-n diameter and center from sensor heights.

Thread crest locations are obtained from the thread region object,keeping only thread crests that are within the inspection region, andthat also have height at least 95% of the median crest height. For avalid inspection region there would then be 5-10 thread crest points persensor for typical usage of the trilobe feature.

To estimate the sensor heights an estimation process that is robustenough to tolerate several invalid thread crests is needed. A preferredprocess uses a robust line fit procedure to obtain a line fit throughthe thread crests that will not be influenced by one or two invalidcrest data items. Once the robust line is found, the sensor heightestimate is the robust line's height at the midpoint of the inspectionregion.

Robust Line Fit Procedure

The robust line fit is a simple parameter sampling process. For everypair of points in the data set to be fit, an evaluation line isproduced. A figure of merit for every evaluation line is produced and isthe RMS distance per point between the data and the evaluation line. TheRMS distances are sorted and the evaluation line with the median RMSdistance is chosen.

This procedure is computationally costly but can work correctly with upto 49% of the data as outliers.

Potential Issues with Trilobe Region Signal Processing

Inspection Region Taper May Bias Results

The estimation process is model-based and the model is a trilobecylinder. Thus, a taper in the threaded region, such as near a threadpoint, would provide data that the model fitting process would not becapable of analyzing accurately.

Trilobe Threaded Region Crests Should Be Accurately Located

The thread region processing that locates the thread crest input datafor the threaded trilobe estimation process is very general and maymisfit crest shapes that do not match the thread region crest model.

Trilobe Parameter Estimation

Trilobe parameter estimation uses the intermediate data products,laser-n diameter and laser-n center, to compute the following trilobeparameters:

Trilobe Parameter Description C Radius of circumscribed orcircumscribing circle (V-anvil micrometer maximum diameter) DOrthographic projection diameter (Diameter with standard calipers) ERadius of inscribed or inscribing circle (V-anvil micrometer minimumdiameter) K Trilobe “out of round” parameter (i.e., degree oflobulation) K = E − D = D − C 2K = E − C Angle Rotation angle (between 0and 60 degrees) Angle = 0 when Trilobe flat is parallel to x-axis, andlowest point in y Angle = 60 degrees when trilobe flat is parallel tox-axis, and highest point in y xCenter, yCenter The centerlinecoordinates of the trilobe shape.

D Parameter

The trilobe D parameter can be estimated as the average of the laser-ndiameter measurements in the four lasers.

All the values should agree, within the margin of sensor errors.

If a perfect trilobe shape gage is measured, the differences between thelaser-n diameters and D are diagnostic of measurement accuracy and bias.The RMS distance between the laser-n diameters and D is a measure ofdiameter measurement uncertainty. The maximum difference between D andlaser-n diameter is a measure of the maximum per sensor diametermeasurement bias.

Iterative Computation of K, Angle, xCenter, yCenter Parameters

The computation of the K, Angle, xCenter, and yCenter parameters usesonly the laser-n center intermediate data product. The four laser-ncenter data items are exactly enough items to compute the four unknowntrilobe parameters, there is no redundancy.

A direct four parameter search process is difficult. The search wassimplified to an iterative two parameter search with the followinganalysis.

If the (xCenter, yCenter) centerline coordinates of the trilobe shapeare known, K, Angle can be estimated with an exhaustive search process,described below. Once there are estimates of D, K, and Angle, there is acomplete description of the trilobe shape.

With the trilobe shape description the different projections of thetrilobe shape onto the left and right sensors can be calculated. Withthe left and right sensor projections of the trilobe shape the laser-ncenter data can be used to estimate the trilobe centerline coordinates,xCenter and yCenter.

Finally, with the trilobe centerline coordinates the origin of thecoordinate system specifying the laser-n center data can be changed sothat the origin of the next set of laser-n center data is at the trilobecenterline coordinate estimate. Then the process is repeated with thetransformed laser-n center data as input. In this process, the K, Anglesearch progress is presented with data that eventually has a centerlinethat is very close to (0,0). At that point, all the trilobe parameters,K, Angle, D, xCenter, and yCenter, are known.

Here is a short description of the process.

Repeat {   Perform K, Angle search.   Determine xCenter, yCenter.  Recenter Laser-n Center coordinate system. } until (centerlinecorrection is very small).

K, Angle Search

The K, Angle search is carried out by exhaustive enumeration. A2-dimensional grid is constructed with one dimension being the possiblediscrete values of K in the interval (0 . . . kmax) and the otherdimension being the possible discrete value of angle in the interval (0. . . 60) degrees. At each grid point K, Angle, xCenter, yCenter areused to calculate the laser-n center values that would have producedthose values and then an RMS distance between the calculated and actuallaser-n center values.

In a preferred implementation, the discrete grid is sized 25×25resulting in 625 K, Angle parameter values and 625 RMS values. Theminimum RMS grid value selects the K, Angle output value.

K, Angle Fine Search

The K, Angle search is increased in precision by a subdivided search. Arectangular region of K, Angle space equal to a 2×2 grid in the originalK, Angle discrete grid is subdivided into a 25×25 grid and searched.

Then the process is repeated a second time, subdividing the fine grid inthe same manner.

Determine xCenter yCenter

Once K and Angle are known a new estimate for the trilobe centerlinecoordinates can be obtained.

(1) Estimate sensor height difference caused by trilobe shape, for allfour lasers. This difference is a function of the difference between thelaser and trilobe shape angles.

ΔH (laser, trilobe)=f(K, Angle−laserAngle).

(2) Correct the sensor height difference for the trilobe contribution.

ΔH (laser)=ΔH (laser, data)−ΔH (laser, trilobe).

(3) Compute the trilobe centerline coordinates by a least squares fit ofthe corrected sensor height differences.

(xCenter, yCenter)=g(ΔH(1), ΔH(2), ΔH(3), ΔH(4)).

Convergence Criteria

The iteration converges when the difference between the estimates of Kfrom the current and the previous iteration is less than a predeterminedparameter (nominal value 0.0001).

Computation of Derived Parameters

The derived parameters can be computed from the estimated parameters Dand K:

C=D−K.

E=D+K.

Code Implementation Notes

These notes are to guide future improvements in the code.

Trilobe Thread Parameter Estimation—Differences from

Standard Thread Processing

Trilobe region thread estimation has some differences from standardthread processing.

Most of the differences arise from the fact that standard threadprocessing uses a cylinder of circular cross section whereas trilobethread processing uses a cylinder with a trilobe cross section.

List of Features

Calibrated Sensor Stage Axis Position Zero—by Analysis of Begin ConeEdge Signal

The scanning optical head system described above produces a sampledimage of the amount of light and shadow in a particular sensor's beam. Asample is produced each 4 μm of stage travel. The absolute stageposition is not precise or repeatable, as also discussed.

In order to make the stage position coordinates refer to a commonphysical position, the sensor signal is analyzed to find the position ofthe step edge that marks where the sensor passes the beginning of thecalibration cone 40 at the cylinder 96.

Once the sensor stage positions are all referenced to the common begincone position, the positions of all other features are repeatable tohigh accuracy from scan to scan.

Calibrated Sensor Height Position Zero—by Analysis of Begin Cone CenterPosition

The calibration process that relates sensor digitized raw signals tosensor heights calibrates the relative sensor blockage between the0.125″ cone minimum diameter and the 1.500″ cone maximum diameter.

After the table relating the raw signals to sensor heights isconstructed, the table is used to compute the center of the cone 0.125″beginning cylinder 96. That position is used as an offset to make thecalibrated sensor heights read out zero at the center of the 0.125″ conecylinder 96. This process establishes a common (x,y) center referencecoordinate for each of the four light planes.

Calibration Cone Design—Measure 3d Alignment to Stage Axis with Analysisof Two Constant Diameter Regions

The calibration cone 40 has two regions of 0.750″ diameter that define acylinder in space that is concentric with the calibration cone's centralaxis. By measuring the position of the 0.750″ cylinder as seen by thesensors, the calibration software determines the alignment of the stageaxis 28 and the calibration cone axis.

Calibration Cone Design—Measure Light Plane Angle with “Multi-Step”Region

The calibration cone's “multi-step” region contains five preciselypositioned mechanical steps. These steps are used to compute the rollangle of the light plane with respect to the calibration cone's centralaxis.

Signal processing software measures the precise location of each of thefive steps. When the light plane has a roll angle with respect to thecalibration cone 40, the difference in position between the steppositions computed from a laser's left and right sensors is proportionalto the sine of the roll angle.

The analysis software uses the data from all five steps in a leastsquare minimization procedure that computes the roll angle.

Calibration Cone Design—Use Continuous Constant Slope Sensor HeightCalibration Region, Allowing Arbitrary Interpolation

Previous experimental designs of calibration cones used stepped edgesfor the purpose of relating raw digitized sensor signals to calibratedsensor heights. If the sensor response varied between the height of twoadjacent steps then the calibration process would not directly measurethe variation and the resulting calibration might make mistakes atintermediate diameters.

The present design provides data at all sensor heights, in the diameterrange 0.125″ to 1.500″.

Light Plane “Layer Cake”—Reduce Cross Talk

As previously mentioned, the four laser light planes are arrangedparallel to the bottom plate 61 of the optical head 12, in a regularlyspaced array of heights. Adjacent laser light planes are preferablyseparated by about 2.5 mm. The arrangement is shown schematically inFIG. 10.

This layered arrangement was chosen specifically to eliminate or reducecross talk between different laser beam lines. For example, light frombeam line-1 might scatter from the surface of the UUT and go into thesensor for beam line-2.

The primary means of interference is due to scattering from cylindersthat are aligned with the stage axis 28, a geometry similar to thegeometry of FIG. 9. (Also See Appendix A.)

When the laser light planes are at different heights, light from laser-2(for example) which is scattered by the UUT, arrives at the sensor forlaser-1 at a height of 2.5 mm relative to the expected light from thelaser-1 light plane. This scattered light can be blocked by a lightplane receiver aperture slit as described in Appendix A with referenceto FIG. 24 (i.e., telecentric apertured stops).

Light Plane Receiver Aperture Slits—Reduce Cross Talk

The light plane receivers 68 each have linear slit apertures, about 1.5mm high, that accept light from its corresponding light plane generator.Each aperture slit is mounted in the optical head 12 at a differentheight, matched to the height of its corresponding light plane. Lightfrom different light plane generators or transmitters 66, scattered bythe UUT, is effectively blocked, thereby increasing measurementaccuracy.

Light Plane Receiver Aperture Pinholes—Reduce Forward Scattered Light

Each light plane receiver 68 includes photodiodes which are each fittedwith circular apertures that make the light plane receiver 68telecentric. This aperture pinhole accepts light rays from the nominalangle of incidence and/or from angles of incidence that are onlyslightly different (<1-2 degrees). This means that light beams thatenter the light plane receiver 68 at larger angles of incidence will beblocked by the pinhole mask and not recorded by the measurementcircuitry. Appendix A describes this.

The pinholes reduce systematic measurement errors caused by shinycylindrical parts. For those parts forward scattered light will tend tosystematically reduce the diameter measured because scattered lightfinds its way into the light plane receiver 68.

Light Plane Generator Module—Alignment Method to Ensure Low BeamDivergence

The Laser Lab measurement system 10 has a requirement that the lightrays from each light plane generator module 66 be parallel and notdivergent.

The apparent diameter of a 12.7 mm cylinder should not change by morethan 0.0025 mm as the cylinder center is moved (+/−) 1 mm from thecenter of the measurement area. This requirement couples a requiredmeasurement accuracy bias (0.0025 mm) with an estimated accuracy of partplacement by customers (+/−1 mm.).

This requirement places limits on the alignment accuracy of the lightrays within the light plane, or its divergence. In the worst case, thebeam through the center of the cylinder is at an angle of zero, the leftshadow ray is at an angle of −φ, and the right shadow ray is at an angleof +φ. This would mean the maximum misalignment angle for any shadow rayin the light plane is less than 1.3 mrad.

These maximum misalignment angles translate to an accuracy of focus whenmanufacturing or assembling the light plane generator module 66. Analign and focus instrument or alignment fixture, generally indicated at100 in FIG. 31, provides a high precision mechanical adjustment of theposition of lens 316 (i.e., FIG. 37) so that the beam divergence isminimized when constructing each light plane generator module 66. Oncethe adjustments are complete, the adjustments can be permanently fixedin place by tightening adjustment screws and by gluing the mechanicalattachment points to prevent movement. A fuller description of thealignment method is provided in Appendix D.

Light Plane Generator—Alignment Method to Ensure High Light BeamFlatness

The Laser Lab measurement system 10 also has a requirement that thelight plane be generally flat. It was discovered that if the opticalelements of the light plane generator module 66 were misaligned then thelight plane's image on a flat target would be curved, rather thanstraight.

A curved light plane would make the light plane-to-calibration coneangle calibration described above invalid. A curved light plane wouldalso make the sensor height calibration described above inaccurate. Thecurve in the light plane would make predicting the diameter of thecalibration cone 40 as a function of stage position much less accurateand make the sensor height calibration much less accurate.

The align and focus method as noted above and as described in Appendix Dis designed to allow the light beam flatness of the light planegenerator module 66 to be effectively minimized during moduleproduction. It was found that the angular and rotational alignment ofthe lens 310, 312 and 316 to the module base plane was an importantvariable. These alignments when performed sequentially allow the lightplane generator module 66 to be setup to meet the flatness requirement,at which point the adjustments are permanently fixed in place bytightening adjustment screws and gluing mechanical attachment points toprevent movement.

Flatness is eliminated primarily by the adjustment of the lens 316, FIG.37, the first cylinder lens. The align and focus instrument contains arotating arm, with a clamp 188, FIG. 31.

The rotating arm's clamp holds the plate 318, FIG. 37, attached to thefirst cylindrical lens 316, during alignment of the transmitter module.

Rotation of the clamp 188 causes the laser line image at the target 210to transition between line shapes on the target of curved upwards, flat,and curved downwards.

A secondary adjustment is by rotating the lens 316. This adjustmentprimarily affects the inclination of the laser line image at the target210, not the curvature. The inclination is adjusted to make the laserline image horizontal.

The adjustment of lenses 316 and 312 is inter-dependent.

Light Plane Receiver—Alignment Method

The light plane receiver modules 68 also have alignment requirements.The optical elements of the modules 68 are precisely positioned so thatthey precisely focus the light from their respective light planegenerator module 66 within the pinhole apertures of the detectors (i.e.,FIG. 24).

The receiver module 60 accepts the light from all possible light rayswithin the light plane at approximately the same efficiency, so thegenerator/receiver subsystem (i.e., 66 and 68, respectively) will have asmooth light acceptance profile, as a function of distance across thelight plane. This is a requirement from the sensor height calibrationprocess.

The light plane receiver modules 68 and the light plane generatormodules 66 are capable of working together when mounted on the opticalhead base plate 61 at standard hole positions.

The receiver light plane split line is centered within the light plane.The align and focus method allows for the proper assembly and subsequenttesting of the light plane receiver module 68 and its components asdescribed in this application.

Referring now to FIGS. 37-44, there is illustrated a common mirror lensmount, generally indicated at 304, of the transmitter modules 66.Preferably, the mount 304 is manufactured from a single piece of lowexpansion al-mag alloy to ensure dimensional stability over a widetemperature range. The mount 304 is precisely machined to ensure thatthe various reference surfaces of the mount 304 are properly positionedwith respect to each other.

FIG. 37 is an exploded perspective view of the mount 304 together withits various supported mirrors 306, a first cylindrical lens 316, asecond cylindrical lens 312 and a third cylindrical lens 310. The mount304 includes an integrally formed lens holder 314 for the secondcylindrical lens 312. The third cylindrical lens 310 is mounted on afront reference surface 311 of the mount 304. The second cylindricallens 312 is held within its holder 314 which is at least partiallydefined by a reference surface 313 as best shown in FIG. 61. The firstcylindrical lens 316 is held within an adjustable mounting assembly,generally indicated at 320, which includes a base plate 319 and a lensmount 318.

A rear one of the mirrors 306 is held within an adjustable mirror mount322 which is mounted at a back reference surface 323 of the mount 304(i.e., FIG. 39). The front one of the mirrors 306 is mounted internallywithin the mount 304 at an internal reference surface 325 as best shownin FIGS. 43 and 39. Comparing FIGS. 39 and 43 (sectional views of themount 304 without and with the mirrors 306 and the lenses 306, 310 and312, respectively), the rear mirror 306 is mounted with respect to theinclined reference surface 323 whereas the internally mounted mirror 306is mounted within the module 304 with reference to the inclinedreference surface 325.

The mirrors 306 are preferably made of BK-7 material whereas the lenses310, 312 and 316 are made of SF-11 material. The lenses 310, 312 and 316are optimized for a laser beam wavelength of 650 nm. Also, the nominalaffective focal lengths for the lenses 310, 312 and 316 are 107 mm, 154mm and 2.75 mm, respectively.

The following sequence of assembly steps for the transmitter module 66are followed, which steps are described in detail in Appendix D:

1. Secure lens 316 to lens mount 318 using a UV adhesive2. Secure lens 312, lens 310 and front mirror 306 to lens and mirrormount 304 using UV adhesive3. Secure back mirror 306 to adjustable mirror mount 322 using UVadhesive4. In sequence:

-   -   position rear mirror 306 mounted to adjustable mirror mount 322        within lens and mirror mount 304 so that a reference laser beam        entering the sub-assembly parallel to the mounting base and at a        height of 0.984″±0.004″ exits the mounting aperture at the front        reference surface 311 of the mount 304 at a height of        1.679″±0.020″ and remains parallel to the mounting base within        0.05°    -   position lens 312 so that the reference laser beam deviates less        than 0.03°    -   position lens 316 rotationally perpendicular to the beam axis so        that the beam is visually flat    -   position lens 316 along the beam axis so that the beam        divergence in the horizontal axis is within ±0.10 m radians    -   position lens 312 rotationally perpendicular to the mounting        surface of lens and mirror mount 304 so that the beam is        parallel to the assembly mounting base within 0.25°    -   position lens 310 so that the reference laser beam is 1.679″        from the assembly mounting surface when measured at a distance        of 10.55″        5. Completed assembly specifications    -   beam tilt perpendicular to laser beam axis—less than 0.50°    -   beam tilt parallel to laser beam axis—less than 0.10°    -   beam height—1.679″±0.020″    -   beam divergence—less than 0.25 m radians        6. Secure all fasteners and adjustable components using epoxy        adhesive.

Referring now to FIGS. 45 and 46 a through 46 d, there is illustrated indetail the laser steering mirrors 62 and their associated componentparts illustrated in FIG. 5. The various relative positions of the lasersteering mirrors 62 as they are adjustably mounted at the top surface ofthe base plate 61 are illustrated in FIGS. 46 a, 46 b, 46 c and 46 dwherein each mirror 62 is mounted at a different angle with respect toits respective translation plate 400. In turn, each of the translationplates 400 is adjustably mounted to the plate 61 at its top surface bymounting screws (FIG. 5). Mounting screws 402 and their associatedwashers 403 secure flanges 404 of their mirror mounts, generallyindicated at 406, to their translation plates 400. In turn, the mirrormounts 406 are located on their plates 400 at different angularpositions, as illustrated in FIGS. 46 a through 46 d. Each plate 400includes an elongated aperture 408 which allows the plates 400 to beadjustably positioned at precise angular positions on the top surface ofthe plate 61. Pins 410 are provided for precise mounting the mirrormounts 406 to their plates 400 and mirror mounting plates 412 to theirrespective mirror mounts 406. Screws 414 and their associated washers416 are provided for securing the mount plates 412 to the mirror mounts406. The mirrors 62 are secured to their mount plates 412 by anadhesive.

Referring now to FIG. 47, which is an exploded perspective view of oneof the light plane receiver modules 68. There is illustrated in FIG. 47a mirror and lens mount, generally indicated at 500. The mirror and lensmount 500 is substantially identical to the mirror and lens mount 304 ofeach of the light plane generator modules 66. The receiver module 68also includes a photodetector mount, generally indicated at 502, a lensmount receiver pair, generally indicated at 504, and a cylindrical lens506, which is substantially identical to the cylindrical lens 312 of thelight plane generator module 66. The receiver module 68 further includesa pair of spherical lenses 508 which are fixably secured at a frontsurface of the lens mount 504 in spaced relationship by an adhesive. Thereceiver module 68 still further includes a pair of circular aperturedelements 510 and a detector PCB assembly mount 512 on which a pair ofphotodetectors 514 are mounted in spaced relationship. The lenses 508typically are designed to operate at a wavelength of 650 nanometers, aremade of SF-11 material and have a nominal affective focal length of 25.8mm.

The photodetector mount 502 includes upper and lower halves 501 and 503,respectively, which are secured together by screws 505.

Typically the lenses 508 are secured to the lens mount 504 with a UVadhesive. Then the lens mount 504 is secured to the mount 500 by screws516 and their associated washers 518, as illustrated in FIG. 5.Typically, the lens 506 is adhesively secured to a front referencesurface of the mount 500, as also illustrated in FIG. 5, by followingthe sequence of steps noted below.

The apertured elements 510 are secured within spaced holes in thephotodetector mount 502 so that the elements 510 are intimate with orimmediately adjacent to the supported detectors 514 and centered withinthe mount 500. The following sequence of assembly steps are followedwhich are described in detail in Appendix D:

-   -   Position lens 506 at the front reference surface of the mount        500 so a reference laser beam entering the subassembly is        parallel to a bottom reference surface of the base of the mount        500 at a height of preferably 1.679″±0.004″ and deviates less        than 0.03°.    -   Position lens 506 rotationally perpendicular to its mounting        surface of the mount 500 so that the reference beam frame is        parallel to the mounting base within 0.25°.    -   Position the receiver lens pair 508 within the mount 500 along        the beam axis so that the energy as measured using a reference        pair of detectors is balanced within 2%; and    -   Position by resulting detector assembly so that the energy        measured using the mounted detectors 514 is balanced within 1%.

After the above-noted steps are performed, all the fasteners andadjusted components are secured using an epoxy adhesive.

The detector PCB assembly mount 512 is secured at the back surface ofthe mount 500 by screws 520 and their respective spacers 522.

LaserLab Out-of-Center Feature

Purpose

This feature is intended to be used to check the placement of a part.The repeatability of measurement of certain features (e.g. thread andtrilobe) depends on how well the part is centered in the machine.

The feature measures the distance from the stage-movement axis,measurement or z-axis to the centerline of a cylinder-symmetrical part,at a specific height.

Use Case

The user selects a section on the part from which he wants to find thedistance to the z-axis. The height where the distance is taken is at themidpoint of this interval. A limit on this distance may then be placed;the default value is 1 mm.

When doing a measurement, if the distance is larger than the limit, amessage box pops up informing the operator that the part is placedbadly. The operator then dismisses the box, and can optionally adjustthe placement of the part and repeat the measurement. The featuremeasurement results will not show up in the table on the main sortscreen, but it is visible in the template editor.

Algorithm

The system first averages the measured height of each of the eightsensors (four lasers, two sensors per laser) over the selected section.

The eight averaged values are used to calculate the x- and y-coordinatesof the center (viewing the stage from front, x points left to right, andy front to back). The distance from the center is the square root of thesum of squared values of x and y.

Details: From FIG. 48, it can easily be deducted that:

$\begin{matrix}{C = {LaserCenter}} \\{= {{RightSensor} - {LeftSensor}}}\end{matrix}$ phi = angle_of_laser x, y = center_of_partC = sin (phi)x − cos (phi)y.

For four lasers, we get four linear equations relating four laser-centerdistances (C1,C2,C3,C4) to two part-center dimensions (x, y). Usinglinear-algebra the equations can be inverted to calculate the centerdimensions:

$C = {{A \times {\begin{bmatrix}x \\y\end{bmatrix}\begin{bmatrix}x \\y\end{bmatrix}}} = {\left( {A^{T}A} \right)^{- 1}A^{T}{C.}}}$

With A a four rows by two columns matrix of sine and negative cosine ofeach laser; and C a four rows vector of laser centers.

Finally,

R=√{square root over (x ² +y ²)}.

LaserLab Tilt-Angle Feature

Purpose

This feature is intended to be used to check the placement of a part.The repeatability of measurement of certain features (e.g. thread andtrilobe) depends on how well the part is aligned with the stage-movement(z-axis).

This feature measures the angle between the centerline of the part andthe z-axis.

Use Case

It is recommended to use the part-tilt check whenever a trilobe orthread feature, or a thread start- or end-anchor is to be measured.

Setup

In the template edit screen, the user adds a new tilt-angle feature, asdetailed in FIG. 49. The user then sets the measurement interval on asection of the part that has the same diameter. The accuracy ofmeasurement depends on the length of the interval; it is recommended tomake the interval-length at least 10 mm.

On this feature the maximum angle (positive tolerance limit) is set to 1degree. The user may change this value.

Run-Time

At run-time, the operator will not see the result of the part-tiltfeature. Instead, if the part is placed such that the angle exceeds thetolerance (more than 1 degree of tilt), the operator is shown a warning.The operator then adjusts (or replaces) the part, dismisses the warningand the part will be inspected again.

Algorithm

From the scanned part, a slice with height of 4 mm is taken near thestart and end of the interval, as illustrated in FIG. 50.

From the averaged sensor-data, each slice's center-point (in (x,y,z)coordinates) can be found with the method as described in theOut-of-Center portion of this application immediately above. Then, a3D-vector from start-slice-center to end-slice-center (Δx,Δy,Δz) isgenerated as the difference between the two slice-centers. The tangentof the tilt-angle is:

${\tan \left( {{tilt}\text{-}{angle}} \right)} = \frac{\sqrt{{\Delta \; x^{2}} + {\Delta \; y^{2}}}}{\Delta \; z}$

The value of 4 mm is chosen to be larger than the maximum pitch-diameterof a threaded part. In this way, the major diameter of a threadedmeasurement interval can be determined. If the measurement intervallength is less than 8 mm, then the height of the slices is reduced tohalf the interval length.

The calculated angle is the average/angle of the part-tilt front start-to end-slice.

Glossary

3-Point distance: Distance from a single point in one laser's sensor toa reference line produced by two points in the laser's other sensor.

3-Crest diameter: Diameter measurement produced from a statisticalaverage or median of 3-point distances.

Align and focus instrument: An optical/mechanical fixture formanufacturing or assembling the light plane generator and light planereceiver modules to the required optical tolerances.

Base/slide unit: The physical base for the Laser Lab measurementhardware. The unit includes a relatively large, heavy triangular baseand a vertical unit containing a motor, a slide, and a linear encoder.

Beam line: A set of optical, mechanical, and electrical componentscontained inside an optical head that create a light plane from a singlelaser beam and convert the shadowed light plane to electrical signalsfor transmission to the PC tower.

Calibrated part data: A data structure consisting of calibrated sensordata from four lasers, each having left and right sensors; may includethe data obtained from scanning the holding devices as well.

Calibrated sensor data: An array of (stage position, height)measurements for one sensor.

Calibration cone: A precisely manufactured single piece of tool steelwith a “cone-shaped” outline. The outline includes cylindrical andfrustum-shaped outlines. The calibration cone or device is used withinthe system to convert sensor digitized raw signals to physicalmeasurements.

Calibration data: The set of tables and parameters that are computedduring the calibration analysis process from sensor raw data. This dataset is also an input to the sensor data calibration process thatconverts sensor raw data to sensor calibrated height data.

Correlation detector: A signal processing algorithm that matches apattern with a vector data set.

Diameter calibration: The process of creating calibration data sets foreach sensor. Each data set allows analysis software to convert thesensor digitized raw signal to a calibrated sensor height. The processis a two stage process including sensor height calibration and finaldiameter correction.

Final diameter correction: A final correction to diameter measurementsbased on a system-level diameter gage pin calibration.

Flank angle: The angle between the thread's flank and the thread's axis.

Flank line—data extraction region: The region containing the central(70%) portion of sensor data for one flank line.

Inspection region: All the calibrated part data between two stageposition limits, start and end positions; includes the holding deviceswhen the holding devices are scanned to determine the geometricmeasurements of the openings in apertured end surfaces of a part.

Intermediate data: Data produced by signal processing of calibratedsensor data that is not saved in the template, but used in estimates ofthread parameters.

Intermediate data—correlation crossing: A match point from thecorrelation between a pattern and (typically) sensor height data.

Intermediate data—flank line: The line fit to data within the flank linedata extraction region.

Intermediate data—crest/root: Measurements of the locations (stage pos,height) of the crests and roots present in the thread form.

Intermediate data—thread 3-D cylinder: Measurement of the 3-D cylinderformed from the least squares fit of the thread crests. The cylinder hasparameters that include its diameter, per point RMS distance betweendata and fit, and cylinder axis.

Intermediate data—wire position: Location of a virtual wire asdetermined from the positions of the left and right adjacent flanklines.

Intermediate data—wire position search interval: Region that containstwo adjacent thread crests, a thread root and two thread flanks.

Laser number: The enumeration of the four lasers in the optical head.The lasers are numbered laser-1 through laser-4.

Light plane generator or transmitter module: Physical module withoptical and mechanical components that converts a light beam generatedby a laser into a plane of light having parallel light rays.

Light plane receiver module: Physical module with optical, mechanical,and electrical components that converts a plane of light having parallellight rays into left and right electrical signals.

Light plane split line: Imaginary line that splits a light plane intoleft and right parts or portions. The line is defined in the light planereceiver module and represents the response to a shadow in the lightplane in the left/right receivers.

Left/right receiver (Rcvr): The receiver is a component of the lightplane receiver module that converts light energy incident on its surface(i.e., image plane) into an electrical signal. The signal current isproportional to the amount of incident light.

Left/Right sensor digitized raw signal: The output of the light planereceiver electronics, after processing of the left or right receiversignals, preferably including current-to-voltage conversion,amplification, analog filtering, and digitization.

Light plane receiver electronics: System module that converts left andright electrical signals from the laser line receiver module intodigitized left and right raw sensor signals, and stores the results inthe PC memory.

Measurement trigger signal: Signal from linear encoder electronicstransmitted to receiver electronics. One measurement trigger signalpulse causes all sensor signals to be sampled and stored.

Optical head: A container such as a sealed metal box containing fourbeam lines and supporting electronics.

Part holder: A mechanical subassembly mounted to the base/slide unit.The subassembly contains the part holder base on which the parts arereceived and retained by the lower holding device while being scanned.The part holder also holds the calibration cone in a stable position soit can also be scanned.

PC tower: The PC tower is a chassis containing a computer, a set ofcontrol electronics modules, and a number of power supplies.

Sensor digitized raw signal: The output after processing of a left/rightreceiver signal by the light plane receiver electronics.

Sensor height: The distance from the laser plane split line to thelight/dark shadow edge of a part, such as an ammunition case, or aholding device, measured in the sensor light plane.

Sensor height calibration: The process of acquiring and analyzing a setof data that is used to convert a sensor digitized raw signal to asensor height.

Sensor number: The enumeration of the eight sensors in the optical head.The values range from laser-1 left (L1L) through laser-4 right (L4R).

Sensor raw data: The set of data generated by the Laser Lab sensorsystem during one scan of the calibration cone and the UUT.

Sensor calibrated height data: The Laser Lab sensor raw data setgenerated from one scan, converted to physical units and corrected forall known issues.

Shadow ray: The ray of light that just grazes the surface of the UUT.

Stage axis: The direction in 3-D space defined by the up/down movementof the optical head's mechanical stage.

Stage axis, sensor zero position: The position of the beginning of thecalibration cone, as determined by analysis of the sensor digitized rawsignal.

Stage axis, calibrated sensor position: The raw position of the stageaxis, corrected to be 0 mm at the sensor zero position. The calibratedsensor positions are different between sensors, even between the leftand right sensors for the same laser.

Stage axis, raw position: The stage axis raw position is the value ofthe linear encoder counter, maintained by the linear encoder electronicsmodule. The zero position is set when the stage stops during a “home”command; the linear encoder electronics module senses a bottom tripsignal, stops the motor, and then zeroes the encoder counter. Thisprocess produces a final resting physical stage position that varies byseveral hundred microns, depending on the length of the move, the stagespeed, and other factors. This physical position is generally toouncertain for direct use in specifying positions on a part or a holdingdevice.

Thread model: An estimate for one cycle of the repeated thread form,learned at template edit time.

Thread parameter—functional diameter: Estimate of the diameter of avirtual nut with the nominal pitch diameter, that could contain theobserved 3-D thread form with all its lead deviations and otherdeviations from perfect form.

Thread parameter—lead deviation: The maximum deviation of the leadposition from the perfect helical form.

Thread parameter—major diameter: The diameter of a cylinder enclosingall the thread crests.

Thread parameter—minor diameter: The diameter of a cylinder through allthe thread roots.

Thread parameter—pitch diameter: The diameter of a cylinder thatintersects the perfect thread form midway between crest and root.

Thread parameter—pitch: The average or median axial distance betweenadjacent threads.

Upper tooling: A mechanical subassembly mounted to the base/slide unit.The upper tooling includes a long stainless steel rod that can be movedup and down to hold a variety of different-sized parts, such asammunition cases. This tooling also includes a spring-loaded part clampor upper holding device that facilitates placement, retention andrelease of parts.

Unit under test (UUT): The part being measured either directly byintersecting planes of laser light or indirectly by planes of laserlight which intersect holding devices which hold the part.

APPENDIX A Split Laser—Diameter Bias Due To Beam Scattering Summary

-   -   scattering of light from a reflective cylinder leads to an        underestimate of its diameter.    -   the scattering from a cone section has a large underestimate        than the cylinder.    -   introducing a pin hole aperture in the laser receiver can limit        this effect, by making the laser receiver telecentric, or        sensitive to light from a narrow range of incident angles.        Scattering from a Perfectly Reflective Cylinder

Scattering of light from a cylinder will cause a systematicunderestimate of the cylinder's diameter in the split laser system.Light incident near the surface of the cylinder is scattered at aglancing angle as illustrated in FIG. 23. If the scattered light getsinto the split laser detector, then the cylinder diameter appearssystematically smaller. The effect is largest if the cylinder isperfectly reflective, and is absent if the cylinder is perfectlyabsorbing. The magnitude of this effect is calculated to set a limit onthe systematic size underestimate.

The Scattering Angle

Assuming a beam of light hits the cylinder and is reflected, the beam'sdirection of travel would reach a depth of ΔH within the cylinder if thebeam direction were continued on a straight line. The beam is deflectedby an angle θ_(refl)=2θ by the perfectly reflecting surface as shown inFIG. 23.

For a cylinder of diameter d the following relationships hold betweenthe scattering angle, θ_(refl)=2θ, the depth ΔH, and the diameter, d:

ΔH=d(1−cos θ)

ΔH≈d(θ²/2)=d(θ_(refl))²/8

θ_(refl)≈2√{square root over (2ΔH/d)}

For ΔH=0.0001″, d=0.5000″, then θ≈2√{square root over (0.0004)}=0.04radian=2.3 degree.For ΔH=0.001″, d=0.5000″, then θ≈2√{square root over (0.004)}=0.13radian=7.2 degree.

Effects on the Diameter Measurement

In the ideal case, with no scattering, the amount of light that reachesthe detector is from line generator light rays that do not intersect thepart or the holding devices. The light signal is then related to theorthographic projection of the part or the holding device, perpendicularto the beam direction.

With scattering, light that would have been blocked may enter thedetector.

The scattered light could fail to enter the detector due to one of thefollowing effects:

-   -   light absorption at the cylinder surface (such as by black        coatings),    -   scattering light may miss the entrance to the laser light        receivers.

An upper limit to the underestimate of light blockage by a perfectlyreflecting cylinder can be calculated. It is assumed that all lightscattered through angles smaller than angle θ_(max) will be received inthe laser light receiver. All light scattered through larger angles islost and is not received in the laser light receiver.

The diameter blockage underestimate is:

Diameter Blockage Underestimate≡2ΔH≈d(θ_(max))²/4

Blockage Underestimate θ_(max) Diameter 0.00005″ 0.02 radian 0.5000″0.00020″ 0.04 radian 0.5000″ 0.00080″ 0.08 radian 0.5000″

Limiting Diameter Measurement Bias—Using Telecentric Ideas at the LaserReceiver

As illustrated in FIG. 24, the simplest model of the laser receiverincludes a focusing lens for each of the left and right sensors, and alaser diode to measure the laser light at each sensor's focusing lens'focal plane. The laser diodes are placed at the focus for a light sourceat infinite distance.

In this model, when a light ray is incident on the focusing lens at adifferent angle, then the light is focused at a slightly different placein the len's focal plane.

The change in position at the focal plane between light incident alongthe optical axis and light incident at an angle θ, is Δpos=ftan(θ)≈f(θ), where f is the focal length of the lens.

A pinhole aperture is added in front of the laser diode to make thelaser receiver sensitive to light only in a small range of incidentangles. Similar apertures are used in the construction of telecentriclenses. The pinhole aperture size is shown in the following table.

pin hole size=2fθ_(max)

Pinhole Diameter (mm) θ_(max) focal length 2 mm 0.01 radian 100 mm  1 mm0.01 radian 50 mm 2 mm 0.02 radian 50 mm 0.5 mm   0.01 radian 25 mm 1 mm0.02 radian 25 mm 2 mm 0.04 radian 25 mm

Light Deflection From a Cone Section

The deflection of light from a reflective cone is also of interest,since conical shapes (i.e., frustums) are utilized in the Laser Labcalibration fixture 40 as illustrated in FIG. 25.

The normal to the cone at the point where the incident beam is justtangent is:

{circumflex over (n)}=[−cos(θ_(cone)),+sin(θ_(cone)),0].

The normal to the cone rotated an angle θ_(rot) about the y-axis (seeFIG. 23):

{circumflex over(n)}=[−cos(θ_(rot))cos(θ_(cone)),+sin(θ_(cone))−sin(θ_(rot))cos(θ_(cone))].

The incident light's direction of travel is:

{circumflex over (l)}=[0,0,1].

The incident light's direction of travel after reflection from a mirroris:

$\begin{matrix}{{\hat{l}}_{refl} = {\hat{l} - {2\; \hat{n}\text{(}{\hat{l} \cdot n}\text{)}} - {2\; \hat{n}\text{(}{\hat{l} \cdot \hat{n}}\text{)}}}} \\{= {2\; {\hat{n}\left( {{\sin \left( \theta_{rot} \right)}{\cos \left( \theta_{cone} \right)}} \right.}}} \\{= {2\; {\sin \left( \theta_{rot} \right)}{{\cos \left( \theta_{cone} \right)}\begin{bmatrix}{{{- \cos}\left( \theta_{rot} \right){\cos \left( \theta_{cone} \right)}},} \\{{+ {\sin \left( \theta_{cone} \right)}} - {{\sin \left( \theta_{rot} \right)}{\cos \left( \theta_{cone} \right)}}}\end{bmatrix}}}} \\{= \begin{bmatrix}\begin{matrix}{{{- 2}\; {\sin \left( \theta_{rot} \right)}{\cos \left( \theta_{rot} \right)}{\cos^{2}\left( \theta_{cone} \right)}},} \\{{{+ 2}{\sin \left( \theta_{rot} \right)}{\cos \left( \theta_{cone} \right)}{\sin \left( \theta_{cone} \right)}},}\end{matrix} \\{{- 2}\; {\sin^{2}\left( \theta_{rot} \right)}{\cos^{2}\left( \theta_{cone} \right)}}\end{bmatrix}} \\{= \begin{bmatrix}\begin{matrix}{{{- {\sin \left( {2\; \theta_{rot}} \right)}}{\cos^{2}\left( \theta_{cone} \right)}},} \\{{{+ {\sin \left( \theta_{rot} \right)}}{\sin \left( {2\; \theta_{cone}} \right)}},}\end{matrix} \\{{- 2}\; {\sin^{2}\left( \theta_{rot} \right)}{\cos^{2}\left( \theta_{cone} \right)}}\end{bmatrix}}\end{matrix}$ $\begin{matrix}{{\hat{l}}_{refl} = {\left\lbrack {0,0,1} \right\rbrack - {2\; \hat{n}\text{(}{\hat{l} \cdot \hat{n}}\text{)}}}} \\{= \begin{bmatrix}\begin{matrix}{{{- {\sin \left( {2\; \theta_{rot}} \right)}}{\cos^{2}\left( \theta_{cone} \right)}},} \\{{{+ {\sin \left( \theta_{rot} \right)}}{\sin \left( {2\; \theta_{cone}} \right)}},}\end{matrix} \\{1 - {2\; {\sin^{2}\left( \theta_{rot} \right)}{\cos^{2}\left( \theta_{cone} \right)}}}\end{bmatrix}} \\{= \begin{bmatrix}\begin{matrix}{{{- {\sin \left( {2\; \theta_{rot}} \right)}}{\cos^{2}\left( \theta_{cone} \right)}},} \\{{{+ {\sin \left( \theta_{rot} \right)}}{\sin \left( {2\; \theta_{cone}} \right)}},}\end{matrix} \\{{\cos \left( {2\; \theta_{rot}} \right)} + {2\; {\sin^{2}\left( \theta_{rot} \right)}{\sin^{2}\left( \theta_{cone} \right)}}}\end{bmatrix}}\end{matrix}$

The angle θ_(refl) between incident and reflected light beams can becomputed.

cos(θ_(refl))=({circumflex over (l)} _(refl) •{circumflex over(l)})=cos(2θ_(rot))+2 sin²(θ_(rot))sin²(θ_(cone)).

For a cylinder, (θ_(cone)=0), the expected result is obtained:

θ_(refl)=2θ_(rot).

For small angles scattering angles, (θ_(rot)<<1), and arbitrary coneangles there is:

cos(θ_(reft))=1−sin²(θ_(rot))cos²(θ_(cone))

1−(θ_(reft))²/2≈1−2(θ_(rot))² cos²(θ_(cone)).

θ_(reft)≈2θ_(rot) cos(θ_(cone))

Practical Considerations in the Laser Lab System

The Laser Lab cone fixture 40 has cone sections (i.e., frustums) withcone angle θ_(cone)=35 degrees and cos(θ_(cone))=0.820. What this meansin practical terms is that the scattering angle on the cone sections isabout 20% less than the scattering angles on the cylindrical sections.The approximate scaling developed in the previous sections indicatesthat the diameter underestimate would then be about 10% more than forthe straight cylinder case.

Energy Absorption on the Cylinder

If the cylinder surface were coated with a light absorbing coating, thenit might be that the reflected light would continue on in the samedirection as the perfectly reflecting case, but with reduced intensity.

A model similar to the one noted above can be developed. In that modelall light scattered between angles 0 and θ_(max) enters the split laserdetector of FIG. 24. Suppose that a fraction f_(absorb) of the scatteredlight, between angles 0 and θ_(max), is absorbed at the cylindersurface, and a fraction (1−f_(absorb)) continues on to the split laserdetector, then the new model is:

2ΔH≈(1−f _(absorb))d(θ_(max))²/4.

Thus, the underestimate is improved by the factor (1−f_(absorb)).

APPENDIX B Summary

This Appendix describes in detail one embodiment of the Laser Labcalibration process.

Table of Contents

General Parameters 104 Geometrical Parameters of the Calibration 104Cone Parameters 104 Boundary Positions - Cone Height Model 105 GeneralParameters of the Calibration 106 Cone Signal Processing 106 Full OpenSignal Level 106 Rough Edge 106 High Precision Edge Positions 107 LaserRoll 107 Data Binning - Cone Slope or Const Diameter Regions 108Calibration Model Analysis 108 Sensor Blockage Table - Cone Slope RegionData 108 Iteration Control 108 Calibration 109 Smooth Cone Description109 Cone Signal Processing 110 Data Partitioning and Consistency Checks110 Rough Step Edge Positions 110 Rough Slope Edge Positions 111 RoughEdge Position Consistency Checks 111 Sensor Data Partition Table 112Full Open Signal Level 112 Full Open Signal Level - Consistency Checks112 High Precision Step Edge Positions 112 High Precision Step EdgePositions - Consistency Checks 113 Begin Cone Sensor Position Offsets114 Laser Roll Computation 114 Consistency Checks 115 Process SensorBlockage Data 115 Identify Cone Slope Data Regions 116 Data Binning 116Data Averaging/Feature Generation 116 Consistency Checks 117 ProcessCone Aspect Angle Data 117 Identify Constant Diameter Regions 117 DataBinning/Averaging/Consistency Checks 117 Calibration Model Analysis 118Sensor Blockage Table and Cone Aspect Angle Calibration 118 ConeProjection onto Sensor 119 Computation of Sensor Blockage Table 119Sensor Blockage Table - Interpolation 121 Sensor Level InterpolationType - Linear 121 Computation of Cone Tilt Angle 121 Step Edge PositionChanges due to Laser Roll Angle 123 Cone 3-D Location Analysis 123Travel Z-Axis to Cone Z-Axis Scaling 125 Iteration Control 125Consistency Checks 125 Analysis and Error Propagation 126 Effect ofSensor Blockage Table Errors due to Roll Angle 126 on Diameter Bias dueto Position Offsets Effect of Roll Angle Errors on Sensor Blockage Table127 Effect of Step Edge Location Errors on Roll Angle 127 Effect ofMultiple Scan Calibration on System Diameter Bias 128 Glossary 128

General Parameters

Geometrical Parameters of the Calibration

For cone parameters, reference drawing FIG. 4 a.

Sampling Interval 0.004 mm Specified by the encoder. (fixed) Theelectronics generates eight encoder trigger pulses, one per each sensor,from each encoder pulse.

Cone Parameters

Start Diameter 3.175 mm Diameter of the face at the beginning of thecone. Const Diameter 19.05 mm Diameter of the two const diameterregions. Max Diameter 38.100 mm Diameter of the cone at the first stepedge. Steps 3.810 mm Height of each step. 1.500 mm Width of each step. 5Number of steps. Inclination Angle 34.005 degrees Cone slope angle.

Boundary Positions—Cone Height Model

Distance Height Description 0.000 0.000 Begin cone, bottom of step 0.0001.587 Begin cone, top of step 3.810 1.587 Begin const diam-0 region Endconst diam-0 region 15.578 9.525 Begin const slope-1 region End constslope-1 region 18.923 9.525 Begin const diam-1 region End const diam-1region 33.020 19.050 Begin const slope-2 region End const slope-2 region35.888 19.050 Begin max diam region End max diam region, begin step-135.888 17.145 End step-1 37.379 17.145 Begin step-2 37.379 15.240 Endstep-2 38.877 15.240 Begin step-3 38.877 13.335 End step-3 40.378 13.335Begin step-4 40.378 11.430 End step-4 41.877 11.430 Begin step-5 41.8779.525 End step-5 51.118 9.525 Begin const diam-2 region End const diam-2region.

General Parameters of the Calibration

Cone Signal Processing

Full Open Signal Level

500 samples RegionSize 0.4 mm GuardRegion ensure full open signal is notcontaminated by scan startup. 100 samples MaxSignalVariance signalvariance >0.2% (100/50000) is diagnostic of poor laser sensor signalquality. 45000 MinSignalLevel 64000 MaxSignalLevel. signal levelsoutside this band are diagnostic of laser adjustment problems.

Rough Edge

0.4 mm Step Size typical beam “sigma” is 15 samples or 60 um. 0.4 mm isabout 6-7 sigma. 0.4 mm Smooth Size typical beam “sigma” is 15 samplesor 60 um. 0.4 mm covers 6-7 sigma. 200 levels Step Threshold value largeenough to minimize false edge detections. −1500 levels Expected BeginCone Step Height. +1000 levels Expected Standard Step Height. 7000levels Standard Step Height Tolerance. 0.5 Cone Slope Threshold units of(Step Size divided by Sampling Interval), or samples. This is athreshold on a double finite difference, using Step Size. 2 ExpectedCone Slope Height. 3 Cone Slope Height Tolerance. units of (Step Sizedivided by Sampling Interval) or samples. 10 mm Expected Begin ConeOffset. 10 mm Begin Cone Step Position Tolerance. 1 mm Standard StepPosition Tolerance. tolerances are for the relative measurement of stepto begin cone step. 1 mm Standard Slope Position Tolerance. tolerancesare for the relative measurement of step to begin cone step.

High Precision Edge Positions

0.1 mm MaxPosDev maximum deviation of position from expected position.3.0 mm MaxHeightDev maximum deviation of step height from expectedvalue. 1000 levels Expected Height. 100 levels MaxRmsDev maximumaveraged RMS(fit-data) for the step edge. 70% LineFitCentralRegionpercentage of central data used for transition line fit. 0.4 mmLineFitRegionSize size of region before and after step used for linefits. 0.4 mm LineFitGuardRegion size of guard region betweenbefore/after step line fit and transition region.

Laser Roll

0.100 mm MaxRmsDeltaStepPos step position delta RMS average larger thanthis value is diagnostic of laser beam or calibration device problems.

Data Binning—Cone Slope or Const Diameter Regions

0.200 mm GuardRegion minimum distance, diameter data bin to regionboundary. 0.200 mm RegionBinSize size of diameter data bin. 50 levelsMaxDataBinSigma maximum data bin order statistic sigma estimator. 25levels MaxDataBinLinDev maximum data bin deviation from neighbor linearestimator. 1 Fit order (0 constant, 1 linear).

Calibration Model Analysis Sensor Blockage Table—Cone Slope Region Data

0.01 mm SensorMaxInterpolationError maximum interpolation error, due toshape of table. “Linear” SensorInterpolation Type specify type of SensorBlockage Table interpolation. “Linear” ZeroHeightExtrapolationMethodmethod to extrapolate Sensor Blockage table. 10ZeroHeightExtrapolationPoints number of points in the extrapolation dataset. “Linear” MaxHeightExtrapolationMethod method to extrapolate SensorBlockage table. 10 MaxHeightExtrapolationPoints number of points in theextrapolation data set.

Iteration Control

4 MaxIterations maximum number of Sensor Blockage Table iterations.

Calibration

Smooth Cone Description

FIG. 26 shows a schematic outline of the sensor signal produced by thesmooth cone. The different regions of the signal are labeled andexplained below.

FIG. 26: Cone Outline—One Sensor's Signal

The signal is plotted with full open sensor level shown at the bottom ofthe figure and fully blocked sensor level shown at the top of thefigure.

Full Open Region—beginning of scan to beginning of cone.

-   -   establish the full open level.    -   establish the exact position of the beginning of the cone.        Cone Slope Data—two constant slope regions separated by a const        diameter step.    -   establish the sensor blockage table, the correspondence between        sensor level and known diameters.        Const Diameter Data—two constant diameter data regions, 19.050        mm diameter.    -   establish the center line of the cone.    -   establish the position of the cone center line, relative to the        stage travel axis.        Steps—Roll Angle Data—five constant height, constant width        steps.    -   establish the angle of the laser light plane, relative to the        cone center line. Support Bracket—no data, all light blocked.        Part Support Cylinder—constant diameter fixed support.    -   top of this cylinder plus height of part support cap may        establish part base position.        Part Support Cap—constant diameter cap.    -   provides base for part.    -   center position may vary slightly, caps are not fixed, there are        several for different types of parts, the insertion is easy for        operators to change.    -   height and diameter are more precise than the center position.

Cone Signal Processing

In this section the processing and analysis are described that occurprior to creating a sensor blockage calibration table and determiningthe cone tilt angle.

In this section, the raw sensor data is processed and yields features inthe (stage position, sensor level) space.

In the Laser Lab system, the conversion between stage position encodercount and stage position is simple and requires no calibration. Thestage encoder count is multiplied by the Sampling Interval (0.004 mm) toproduce the stage position.

In this section stage positions may be specified interchangeably usingencoder counts or position values.

Data Partitioning and Consistency Checks

The first step is to partition the data into regions for more detailedprocessing. With the smooth cone calibration design the partition can beaccomplished with a combination of the positions of the step edges andknown positions of features on the calibration cone.

Rough Step Edge Positions

The following step edge positions should be identified in the data. Theedges will be identified with a low precision edge finder, using finitedifference detection, with smoothing.

Step Description Beginning of Cone Required. Roll Angle Steps 1 through5 ″

Rough Slope Edge Positions

The following slope edge (2-nd derivative) positions should beidentified in the data. The edges will be identified with a lowprecision edge finder, using finite difference detection, withsmoothing.

Step Description Beginning of Const Diam-1 Required. End of Const Diam-1″ Beginning of Const Diam-2 Required. End of Const Diam-2 ″

Rough Edge Position Consistency Checks

C-1: begin cone step edge exists as first step edge near expectedposition.

“near” has large tolerance, Begin Cone Step Position Tolerance.

C-2: required step edges exist near expected positions.

Expected positions are relative to begin cone step edge.

“near” has moderate tolerance, Standard Step Position Tolerance.

C-2: no other significant edge steps exist in calibration cone regions.

The part region may also have step edges.

C-3: required slope edges exist near expected positions.

Expected positions are relative to begin cone step edge.

“near” has moderate tolerance, Standard Slope Position Tolerance.

If either of checks 1, 2, 3 fail, the calibration process is stopped anddiagnostic and logging messages are generated on the computer or PC.

Sensor Data Partition Table

The sensor data partition table gives the rough stage position ofboundaries between the calibration data regions, on a per sensor basis.The table is used by downstream functions to provide rough startingpoints for find location modules.

Full Open Signal Level

The full open signal level is computed from data in the full openestimation region, shown in FIG. 27. The region's size is specified aswell as the size of two guard regions, for begin part and for beginscan.

The median and the order statistic corresponding to sigma are computed,from the sample data within the full open estimation region.

Full Open Signal Level—Consistency Checks

C-1: The full open estimation region must fit between the two guardregions.C-2: The “sigma” order statistic must be less than the MaxSignalVarianceparameter.

If these checks fail, the calibration fails and diagnostic and loggingmessages are generated on the computer or PC.

High Precision Step Edge Positions

High precision step edge position processing uses the rough edge steppositions as initial locations to find high precision edge parametersfor seven step edges. The step edges are the begin part edge, the partsupport cylinder end step, and the five steps in the roll angle data.For each step edge, four parameters are computed, the step position,step height, the beam width, and a step quality measure.

The high precision edge detector uses three line fits to the step edgedata, one before the step edge, one after the step edge, and one in thestep edge transition region. A fixed size guard region,LineFitRegionSize, keeps non-linear data out of the before/after stepline fit regions. The central LineFitCentralRegion percent of thetransition region data is used for the transition region line fit. FIG.28 shows these data regions.

The fit degree in the before/after part regions can be adjusted to beeither const (degree-1) or linear (degree-2). For example, the begincone step edge requires before step fit degree-1 and after step fitdegree-2. The first step edge requires before step fit degree-2 andafter step fit degree-1. Steps 2 . . . 5 require both before/after stepfits to be degree-1.

For Each Step a Feature is Generated Containing:

-   -   transition region midpoint {position, sensor level}=(Up+Lp)/2.    -   beam width estimator=0.4 (Step Size)/(transition region slope).    -   before/after part line fit parameters.    -   step height=sensor level (Up)−sensor level (Lp).    -   RMS (data-line): per sample value, before part/transition/after        part.    -   RMS (data-line) average=quadrature average of before        part/transition/after part RMS (data-line) values.

High Precision Step Edge Positions—Consistency Checks

C-1: all high precision step edges found, and within MaxPosDev distanceof expected positions.C-2: all step edges have heights within MaxHeightDev of expectedheights.C-3: the RMS (data-fit) average is less than MaxRmsDev.

If these checks fail, the calibration fails, diagnostic and loggingmessages are generated on the computer or PC.

Begin Cone Sensor Position Offsets

The set of high precision step edge positions at the begin cone stepedge defines a position offset for each sensor.

The position scale is defined separately for each laser sensor. Theposition offset defines the 0-position for feature processing.

Laser Roll Computation

Laser roll is computed from the high precision step edge positions foundin the roll angle data region.

The measurements and parameters utilized are the following.

StepPos(laser,sensor,i)=high precision step edge position, for i-thstep.

Diam(i)=diameter of i-th step, defined to the midpoint of the step.

The difference between StepPos(laser, Left, i) and StepPos(laser, Right,i)

ΔstepPos(laser, i)=StepPos(laser,Right,i)−StepPos(laser,Left,i),

is related to the roll angle of the laser line, β. β is positive whenthe Left sensor edge position is less than the Right sensor edgeposition. β would be viewed as a counter clockwise rotation if the leftsensor height is plotted as a positive number and the right sensorheight is plotted as a negative number:

ΔstepPos(laser,i)=sin(β)Diam(i).

The five equations relating ΔstepPos(laser,i) to Diam(i), can beexpressed as the matrix vector equation below:

A sin(β)=b.

The data can be reduced with a least squares solution of the matrixvector equation, producing a single parameter estimator, β_(est)

sin(β_(est))=(A ^(T) A)⁻¹(A ^(T) b),

where

${b = \begin{pmatrix}{\Delta \; {{StepPos}\left( {{laser},1} \right)}} \\\; \\\ldots \\\; \\{\Delta \; {{StepPos}\left( {{laser},5} \right)}}\end{pmatrix}},{A = \begin{pmatrix}{{Diam}(1)} \\\; \\\ldots \\\; \\{{Diam}(5)}\end{pmatrix}}$${A^{T}A} = {\sum\limits_{i}\left( {{Diam}(i)} \right)^{2}}$${A^{T}b} = {\sum\limits_{i}{\Delta \; {{StepPos}\left( {{laser},i} \right)}{{Diam}(i)}}}$

Consistency Checks

C-1: Verify that the RMS distance per data point between the data,{ΔstepPos(laser,i)}, and the values predicted from the fit,{sin(β_(est))Diam(i)}, is less than MaxDeltaStepPosVariance.

Process Sensor Blockage Data

Raw sensor readings are processed in the cone slope region to produce aset of (sensor level, stage position) features. Typically each featureis based on a small region (10-500 samples) of data. The features areonly generated in regions of valid data, for example they are kept awayfrom step edges by a guard region.

Identify Cone Slope Data Regions

The first cone slope data region lies between the begin cone step edgeand the beginning of a const diameter data angle region. The secondregion starts at the end of the same const diameter cone aspect angleregion and extends to the position of the first roll angle step.

Approximate per sensor boundaries of the cone slope data regions areavailable from the Sensor Data Partition Table, computed in DataPartitioning and Consistency Checks step.

Data Binning

Data is only binned further than GuardRegion counts from the cone sloperegion boundaries, to prevent systematic diameter calibration offsets.

The regions are divided into the number of bins specified by theRegionBinSize and the number of sample positions in the cone sloperegion. If the number of available sample positions is not divided byRegionBinSize, then extra samples are added to the guard regions.

The data bins are not overlapping.

Data Averaging/Feature Generation

Sensor data observations within the bin are processed, forming estimatesof sensor levels and variance within the bin.

Position data within the bin are processed to form a bin positionaverage.

The data within a bin are fit to a first order linear model.

For each data bin, a feature is generated, containing:

-   -   bin position average.    -   bin const, first order linear fit terms.    -   bin fit value at bin average position.    -   bin RMS(fit—data) estimate.

Consistency Checks

C-1: Each data bin's RMS(fit—data) estimator should be less thanMaxDataBinSigma.

C-2: Deviation of each data bin's fit value from the value predicted bylinear estimation using two adjacent bins (left and right) should beless than MaxDataBinLinDev. The end bins are tested using extrapolationfrom two left or two right bins.

Process Cone Aspect Angle Data

The cone's const diameter regions are processed similarly to the coneslope regions.

Identify Constant Diameter Regions

Two constant diameter data regions are processed, see FIG. 27, using theSensor Data Partition Table.

Data Binning/Averaging/Consistency Checks

Data binning, averaging, feature generation, and consistency checks arethe same as for the diameter calibration data regions.

Calibration Model Analysis

Sensor Blockage Table and Cone Aspect Angle Calibration

Cone signal processing produces a laser roll angle estimate and tablesof sensor data at specified stage positions, for cone slope and coneconstant diameter regions.

What is actually required is the projection of the cone 40 onto thelight beam, as a function of the stage position. This projection dependson two angles, the laser roll angle β, and the cone tilt angle α, theangle between the stage travel axis and the cone symmetry axis.

Each of the angles is defined per laser, the full set of laser and coneangles is {B_(i), α_(i)}, where the laser index-i is in the interval (1. . . 4).

The laser roll angle is known, but the cone tilt angle must becalculated. Since the cone tilt angle is small, typically less than1-degree, an iterative process can be successfully defined.

Initially, it is assumed that cone tilt angle α is equal to zero. Withthis assumption, the cone model can be used to generate the expectedprojection of the cone onto the sensor as a function of stage position.The set of cone projections paired with corresponding sensor responsesis used to make the Sensor Blockage Table. After construction, the tablegives the amount of material blocking the sensor, as a function of theexpected sensor response.

The sensor blockage table is then used to process the constant diameterregion data, producing an estimate of the cone tilt angle α.

With the cone tilt angle estimate α, the expected cone projections ontothe sensor and the sensor blockage table are recalculated.

The process is repeated until there is negligible change in the conetilt angle and the sensor blockage table estimates.

Cone Projection onto Sensor

FIG. 29 illustrates cone projection geometry wherein bold lines indicatethe cone outline and wherein:

H₁ is the radius of the start of the cone.

H_(N) is the radius at distance-N along the cone axis.

β is the laser roll angle.

α is the cone tilt angle.

γ is the inclination angle of the cone.

Δp is the position offset due to laser roll.

δ is the change in radius due to laser roll.

Using the spatial relationships shown in FIG. 29, the change in radiusdue to laser roll, δ, is calculated:

(ΔH−δ) tan (β)=Δp

Δp tan(γ+α)=δ

ΔH tan(β)=Δp tan(γ+α) tan(β)+Δp

Δp=ΔH tan(β)/(1+tan(γ+α) tan(β))

Finally, the change in projection is obtained as a function of the laserroll angle, the cone inclination angle, and the projected cone aspectangle.

$\delta = {\Delta \; H\; {\tan (\beta)}\left( \frac{\tan \left( {\gamma + \alpha} \right)}{1 + {{\tan \left( {\gamma + \alpha} \right)}{\tan (\beta)}}} \right)}$

Computation of Sensor Blockage Table

Signal processing of the cone slope data regions produces a list offeatures, one for each sensor data bin. The feature specifies the stageposition and the average sensor level within the data bin:

{SensorBlockage_(i)}={StagePosition_(i), SensorLevel_(i)}.

The {SensorBlockage_(i)} feature is processed to create a calibratedsensor blockage feature:

{CalibratedSensorBlockage_(i)}={SensorLevel_(i),StagePositionOffset_(i), SensorHeight_(i)}.

The StagePosition of the SensorBlockage data bin and the StagePositionof the begin cone step edge are used to compute the stage positionoffset from the beginning of the cone:

StagePositionOffset_(i)=StagePosition_(i)−StagePosition(Begin Cone).

Then the height difference between the begin cone step edge and thestage position is computed:

Δhi=ConeHeight(StagePositionOffset_(i))−ConeHeight(BeginCone).

The correction to ΔH_(i) uses estimates of angles α, β and γ to computethe change in the cone projection, as seen by the sensor:

δ_(i)=δ_(i)(ΔH _(i), α, β, γ),

SensorHeight_(i) =ΔH _(i)−δ_(i).

The following steps summarize the computation:

-   -   sort {SensorBlockage_(i)} features by increasing StagePosition.    -   compute {CalibratedSensorBlockage_(i)}.    -   install the {CalibratedSensorBlockage_(i)} features in the        Sensor Blockage Table.    -   install min,max SensorLevel limits in Sensor Blockage Table.    -   compute interpolation error estimates.

Sensor Blockage Table—Interpolation

The Sensor Blockage Table is analyzed by an interpolating function. Theinterpolating function calculates CalibratedSensorBlockage as a functionof SensorLevel.

Min,max SensorLevel limits are used to specify the region of valid inputfor the interpolation function.

Configuration file parameter SensorLevellnterpolationType is used toselect different interpolation methods at run time.

Sensor Level Interpolation Type—Linear

Linear nearest neighbor interpolation.

The interpolation error estimate compares the sensor height,SensorHeight_(i)=SH_(i) at stage position offset,StagePositionOffset_(i)=SP_(i), with the interpolated sensor heightderived from positions (i+1) and (i−1).

InterpolationErrorEst_(i)=(SH _(i+1) −SH _(i−1))(SP _(i) −SP _(i−1))/(SP_(i+1) −SP _(i−1))−SH _(i,)

InterpolationErrorEst_(i) =ηSH _(i+1)+(1−η)SH _(i−1) −SH _(i), where

η=(SP _(i) −Sp _(i−1))/SP _(i+) −Sp _(i−1))

Computation of Cone Tilt Angle

Signal processing of the cone's constant diameter data regions producesa list of features, one for each sensor data bin. The feature specifiesthe stage position and the average sensor level within the data bin:

{SensorBlockage_(i)}={StagePosition_(i), SensorLevel_(i)].

The Sensor Blockage Table is used to compute features from SB features.

The calibrated constant data region features are used to compute alinear fit to the sensor height data as a function ofStagePositionOffset_(i)=SP_(i).

Left and Right sensor data are fit simultaneously for each laser, sincethe cone tilt angle α affects both. Tilt angle α positive causes thesensor heights to increase in the Left sensor and decrease in the Rightsensor.

${\begin{pmatrix}{+ {SP}_{1}^{L}} & 1 & 0 \\\ldots & \ldots & \ldots \\{+ {SP}_{NL}^{L}} & 1 & 0 \\{- {SP}_{1}^{R}} & 0 & 1 \\\ldots & \ldots & \ldots \\{- {SP}_{NR}^{R}} & 0 & 1\end{pmatrix}\begin{pmatrix}{\tan (\alpha)} \\{Offset}^{L} \\{Offset}^{R}\end{pmatrix}} = \begin{pmatrix}{SH}_{1}^{L} \\\ldots \\{SH}_{NL}^{L} \\{SH}_{1}^{R} \\\ldots \\{SH}_{NR}^{R}\end{pmatrix}$

This equation can be expressed in matrix form:

${A\begin{pmatrix}{\tan (\alpha)} \\{Offset}^{L} \\{Offset}^{R}\end{pmatrix}} = {b.}$

The least squares solution is:

$\begin{pmatrix}{\tan (\alpha)} \\{Offset}^{L} \\{Offset}^{R}\end{pmatrix} = {\left( {A^{T}A} \right)^{- 1}A^{T}{b.}}$

The A^(T)A and Atb have a simple form:

${A^{T}A} = \begin{pmatrix}{\begin{bmatrix}{{\sum\limits_{j}\left( {SP}_{j}^{L} \right)^{2}} +} \\{\sum\limits_{k}\left( {SP}_{k}^{R} \right)^{2}}\end{bmatrix} + {\sum\limits_{j}{SP}_{j}^{L}} - {\sum\limits_{k}{SP}_{k}^{R}}} & \; & \; \\{+ {\sum\limits_{j}\left( {SP}_{j}^{L} \right)}} & \left\lbrack {N_{L} = \underset{j}{\sum 1}} \right\rbrack & 0 \\{- {\sum\limits_{k}{SP}_{k}^{R}}} & 0 & \left\lbrack {N_{R} = {\sum\limits_{k}1}} \right\rbrack\end{pmatrix}$ ${A^{T}b} = \begin{pmatrix}{{+ {\sum\limits_{j}\left( {{SP}_{j}^{L}{SH}_{j}^{L}} \right)}} - {\sum\limits_{k}\left( {{SP}_{k}^{R}{SH}_{k}^{R}} \right)}} \\{+ {\sum\limits_{j}{SH}_{j}^{L}}} \\{+ {\sum\limits_{k}{SH}_{k}^{R}}}\end{pmatrix}$

Step Edge Position Changes due to Laser Roll Angle

The laser roll angle widens the step profile, and also biases theposition.

Cone 3-D Location Analysis

The 3-D cone direction unit vector, {right arrow over (α)}, is observedin each of the 4 laser systems as the cone tilt angle. The projection ofthe cone unit vector, {right arrow over (α)}, into the laser system “I”is (α_(ix), α_(iy), α_(iz)).

In FIG. 30, α₁,α₂,α₃,α₄ are the projections of {right arrow over (α)} onthe y′-axis for the θ=22.5, 67.5, 112.5, 157.5 degree laser sensorsystems. The positive y′-axis is the Right sensor and the negativey′-axis is the Left sensor direction.

The x,y components of the cone unit vector, {right arrow over (α)}, areprojected into laser system “i” with the following equation (the zcomponent along the stage axis is unchanged):

$\begin{pmatrix}\alpha_{nx} \\\alpha_{ny}\end{pmatrix} = {\begin{pmatrix}{{+ \cos}\; \theta_{n}} & {{+ \sin}\; \theta_{n}} \\{{- \sin}\; \theta_{n}} & {{+ \cos}\; \theta_{n}}\end{pmatrix}\begin{pmatrix}\alpha_{x} \\\alpha_{y}\end{pmatrix}}$

The “y” relation is the only one used, since only the α_(iy), α_(iz)components are measured, the sensor does not measure α_(ix).

αny=−αx sin θ_(n)+α_(y) cos θ_(n).

Using measurements in all four laser systems, the two components(α_(x),α_(y)) of the cone unit vector, {right arrow over (α)} areobtained, by solving the following linear equation:

$\begin{pmatrix}\alpha_{1\; y} \\\alpha_{2\; y} \\\alpha_{3\; y} \\\alpha_{4\; y}\end{pmatrix} = {\begin{pmatrix}{{- \sin}\; \theta_{1}} & {{+ \cos}\; \theta_{1}} \\{{- \sin}\; \theta_{2}} & {{+ \cos}\; \theta_{2}} \\{{- \sin}\; \theta_{3}} & {{+ \cos}\; \theta_{3}} \\{{- \sin}\; \theta_{4}} & {{+ \cos}\; \theta_{4}}\end{pmatrix}{\begin{pmatrix}\alpha_{x} \\\alpha_{y}\end{pmatrix}.}}$

These equations can be solved by the least squares method:

$\begin{pmatrix}\alpha_{x} \\\alpha_{y}\end{pmatrix} = {\left( {A^{T}A} \right)^{- 1}A^{T}b}$

where

${A^{T}A} = \begin{pmatrix}{+ {\sum\limits_{i}\left( {\sin \; \theta_{i}} \right)^{2}}} & {- {\sum\limits_{i}{\sin \; \theta_{i}\cos \; \theta_{i}}}} \\{- {\sum\limits_{i}{\sin \; \theta_{i}\cos \; \theta_{i}}}} & {+ {\sum\limits_{i}\left( {\cos \; \theta_{i}} \right)^{2}}}\end{pmatrix}$ ${A^{T}b} = \begin{pmatrix}{- {\sum\limits_{i}{\alpha_{iy}\sin \; \theta_{i}}}} \\{+ {\sum\limits_{i}{\alpha_{iy}\cos \; \theta_{i}}}}\end{pmatrix}$

Travel Z-Axis to Cone Z-Axis Scaling

The distance along the travel axis is different from the distance alongthe cone axis, to second order in the angle between the cone axis andthe travel axis.

${\Delta \; {{pos}\left( {{cone} - {axis}} \right)}} = \frac{\Delta \; {{pos}\left( {{travel} - {axis}} \right)}}{\left( {\alpha_{z} = \sqrt{1 - \alpha_{x}^{2} - \alpha_{y^{2}}}} \right)}$

Iteration Control

Calibration model analysis is an iterative process. The number ofiterations computed is MaxIterations. After computing the last iterationthe Iteration Control stopping criteria is evaluated to determine if avalid Sensor Blockage Table was constructed. (See consistency checks C-3and C-4 below.)

Consistency Checks

C-1: Sorting {SensorBlockage_(i)} features by increasing StagePositionshould produce a SensorLevel list sorted in decreasing order. Thisensures that a Sensor Blockage Table can be produced that predicts aunique sensor blocking height at every valid SensorLevel.

C-2: The maximum absolute value of the interpolation error is less thanSensorLevelMaxInterpolationError.

C-3: The number of iterations is less than or equal to MaxIterations.

C-4: The change in Sensor Blockage Table projection between the last twoiterations Δδ_(i)=δ_(i) ^((n))−δ_(i) ^((n−1)) is less thanMaxSensorProjectionChange.

C-5: The maximum absolute deviation of {SensorHeight_(i)} from the valuepredicted in the cone angle fit is less than MaxRmsHeightDev.

C-6: The maximum absolute deviation of the four cone tilt angles {α_(i)}from values predicted by the fit to the 3-D unit vector {right arrowover (α)} is less than MaxRmsConeAngleDev.

Analysis and Error Propagation

In this section, it can be seen how known errors in the elementary dataitems, such as step edge positions and median sensor values, affect thesystem measurements.

Effect of Sensor Blockage Table Errors due to Roll Angle on DiameterBias due to Position Offsets

Suppose that during calibration there was a roll angle error. Then theleft sensor actual height would be overestimated and the right sensoractual height would be underestimated and systematic diametermeasurement errors would occur. In this situation, the placement offsetof the center of a cylindrical object from the center of the calibrationaxis would cause a systematic offset in the measured diameter.

Roll angle that is too large causes a underestimate of the correctprojection of the cone for the left sensor and an overestimate for theright sensor.

H _(L) =H _(L) ⁰(1−ε)

H _(R) =H _(R) ⁰(1+ε)

The measured diameter can then be shown to have a systematic offset thatis proportional to the roll angle error.

D=H _(L) +H _(R) =H _(L) ⁰(1−ε)+H _(R) ⁰(1+ε)

D=(H _(L) ⁰ +H _(R) ⁰)−(H _(L) ⁰ −H _(R) ⁰)ε

D=D⁰+ε(CtrPosition), where the center position is defined as

CtrPosition=(H _(L) ⁰ −H _(R) ⁰).

The relative error in the diameter measurement, due to the roll angleerror is:

$\frac{\delta \; D}{D}{\delta \left( {\tan (\beta)} \right)}\tan \; (\gamma){\left( \frac{CtrPosition}{D} \right).}$

Effect of Roll Angle Errors on Sensor Blockage Table

Roll angle errors couple with the cone slope to systematically (example)overestimate the left sensor actual height and underestimate the rightsensor actual height, in the Sensor Blockage Table.

H _(L) =H _(L) ⁰(1−tan(β) tan(γ))

For a small roll angle:

δH=±H ⁰ tan(γ) δ(tan(β))

For H=0.500″, γ35°, and δH=0.0001″, δβ≦0.3 mrad=0.016 degree isobtained.

Effect of Step Edge Location Errors on Roll Angle

A simple method to find the roll angle finds the position of two stepedges, and computes the angle from the difference in step positions.

tan(β)=(ΔStepPos)/Diameter

δ(tan(β))=δ(ΔStepPos)/Diameter

The difference in two uncorrelated step positions has approximately 40%greater uncertainty than a single step position.

δ(tan(β))=δ(StepPos)√{square root over (2)}/Diameter

Step positions have σ≈0.005 mm, and at a Diameter=1.400″, this works outto σ(tan(β))≈0.2 mrad.

Effect of Multiple Scan Calibration on System Diameter Bias

The system diameter bias has an error distribution that is similar tothe repeatability distribution for a diameter measurement.

Multiple scans for calibration reduces the bias.

Glossary

Cone tilt angles {α_(i)}: 3-D cone tilt angle projected into eachlaser's coordinate system. 3-D cone tilt angle α: Symmetry axis of conerelative to stage travel axis. Laser roll angle β: Angle of laser lightplane, relative to cone symmetry axis. Sensor Blockage Table: Tablegiving size of object blocking the sensor, as a function of the rawsensor signal.

APPENDIX C

In general, this appendix describes how to fit a cylinder to a set ofpoints. The set of points could be determined in any manner. Theapplication within Laser Lab is the fitting of a cylinder to the set ofthread crest locations. This cylinder is used to estimate the threadregion's major diameter. The data for a thread crest cylindermeasurement is a set of (stage z coordinate, sensor height) pairs. Thesedata points are the observed locations of the thread crests. For a6-pitch thread inspection region, the number of data points per threadis 4 (lasers)*2 (sensors)*2 (flanks)*6 (pitches)=96 (data points).

All the data points are fitted to a simple linear model of the threadcrest cylinder in space with five free parameters:

the cylinder radius, R.

the slope of the cylinder center line, in x and y, (a_(x), a_(y)).

the (x,y) center of the cylinder center line, at the beginning of theinspection interval, (b_(x), b_(y))

The eight data sets are:

{(z_(i)(l, s), h_(i)(l, s))}, wherel=laser label, varying from 1 . . . 4,s=sensor label, L or R, andi=data index, varying from 1 . . . N(l,s).

For example, the data set for the laser-2, right sensor is {(z_(i)(2,R), h_(i)(2, R))}.

A linear matrix equation can be developed to fit a single laser's data.

The parameters are:a(l)=slope of cylinder line in laser-l coordinates.b(=intercept of cylinder line in laser-l coordinates.r=radius of cylinder.

$\begin{matrix}{{{Z(l)}\begin{pmatrix}{a(l)} \\{b(l)}\end{pmatrix}} = {\begin{pmatrix}{z_{1}\left( {l,L} \right)} & 1 \\\; & \ldots \\{z_{N{({l,L})}}\left( {l,L} \right)} & 1 \\{z_{1}\left( {l,R} \right)} & 1 \\\; & \ldots \\{z_{N{({l,R})}}\left( {l,R} \right)} & 1\end{pmatrix}\begin{pmatrix}{a(l)} \\{b(l)}\end{pmatrix}}} \\{= {\begin{pmatrix}{h_{1}\left( {l,L} \right)} \\\ldots \\{h_{N{({l,L})}}\left( {l,L} \right)} \\{h_{1}\left( {l,R} \right)} \\\ldots \\{h_{N{({l,R})}}\left( {l,R} \right)}\end{pmatrix} + {\begin{pmatrix}1 \\\ldots \\1 \\{- 1} \\\ldots \\{- 1}\end{pmatrix}r}}}\end{matrix}$

The one laser equation can be expressed in block form.

${{{\mathbb{Z}}(l)}\begin{pmatrix}{a(l)} \\{b(l)}\end{pmatrix}} = {{(l)} + {{(l)}{r.}}}$

The new vector s(l), is a column of +/−1 values, with +1 for the leftsensor values and −1 for the right sensor values.

${(l)} = \begin{pmatrix}1_{1} \\\ldots \\1_{N{({l,L})}} \\{- 1_{1}} \\\ldots \\{- 1_{N{({l,R})}}}\end{pmatrix}$

One can develop a block matrix equation for a fit of all four lasers'data to nine parameters. The parameters are:

a(1), a(2), a(3), a(4),b(1), b(2), b(3), b(4),r.

The radius parameter is shared by all four lasers, reducing theparameter count from twelve to nine.

$\begin{matrix}{{{\mathbb{Z}}\begin{pmatrix}{a(1)} \\{b(1)} \\{a(2)} \\{b(2)} \\{a(3)} \\{b(3)} \\{a(4)} \\{b(4)} \\r\end{pmatrix}} = {\begin{pmatrix}{{\mathbb{Z}}(1)} & 0 & 0 & 0 & {(1)} \\0 & {{\mathbb{Z}}(2)} & 0 & 0 & {(2)} \\0 & 0 & {{\mathbb{Z}}(3)} & 0 & {(3)} \\0 & 0 & 0 & {{\mathbb{Z}}(4)} & {(4)}\end{pmatrix}\begin{pmatrix}{a(1)} \\{b(1)} \\{a(2)} \\{b(2)} \\{a(3)} \\{b(3)} \\{a(4)} \\{b(4)} \\r\end{pmatrix}}} \\{= \begin{pmatrix}{(1)} \\{(2)} \\{(3)} \\{(4)}\end{pmatrix}} \\{= }\end{matrix}$

Note: Z specifies the matrix containing data from all four lasers, andZ(l) specifies the data matrix containing data from just one laser.

The new four laser equation can be solved by standard least squarestechniques. We will show the solution to develop the structure of theZ^(T)Z and Z^(T)H matrices. We do not actually solve for this set ofnine parameters in practice. In the next section, we will transform theequation to eliminate the dependencies among the a(l) and b(l)parameters, and reduce the number of unknown parameters to five.

ℤ(a(1) b(1)a(2)b(2)a(3)b(3)a(4)b(4)r)^(T) = (a(1) b(1)a(2)b(2)a(3)b(3)a(4)b(4)r)^(T) = (ℤ^(T)ℤ)⁻¹(ℤ^(T))$\left( {{\mathbb{Z}}^{T}{\mathbb{Z}}} \right) = \begin{pmatrix}{{{\mathbb{Z}}^{T}(1)}{{\mathbb{Z}}(1)}} & 0 & 0 & 0 & {{{\mathbb{Z}}^{T}(1)}{(1)}} \\0 & {{{\mathbb{Z}}^{T}(2)}{{\mathbb{Z}}(2)}} & 0 & 0 & {{{\mathbb{Z}}^{T}(2)}{(2)}} \\0 & 0 & {{{\mathbb{Z}}^{T}(3)}{{\mathbb{Z}}(3)}} & 0 & {{{\mathbb{Z}}^{T}(3)}{(3)}} \\0 & 0 & 0 & {{{\mathbb{Z}}^{T}(4)}{{\mathbb{Z}}(4)}} & {{{\mathbb{Z}}^{T}(4)}{(4)}} \\\left( {{{\mathbb{Z}}^{T}(1)}{(1)}} \right)^{T} & \left( {{{\mathbb{Z}}^{T}(2)}{(2)}} \right)^{T} & \left( {{{\mathbb{Z}}^{T}(3)}{(3)}} \right)^{T} & \left( {{{\mathbb{Z}}^{T}(4)}{(4)}} \right)^{T} & {\sum\limits_{l = 1}^{4}{{^{T}(l)}{(l)}}}\end{pmatrix}$ ${{\mathbb{Z}}^{T}H} = \begin{pmatrix}{{{\mathbb{Z}}^{T}(1)}{(1)}} \\{{{\mathbb{Z}}^{T}(2)}{(2)}} \\{{{\mathbb{Z}}^{T}(3)}{(3)}} \\{{{\mathbb{Z}}^{T}(4)}{(3)}} \\{\sum\limits_{l = 1}^{4}{{^{T}(l)}{(l)}}}\end{pmatrix}$

One now determines the numerical results for the important submatrices.

First, the Z^(T)(l)Z(l) submatrix for laser l.

${{{\mathbb{Z}}^{T}(l)}{{\mathbb{Z}}(l)}} = \begin{pmatrix}{{\sum\limits_{i = 1}^{N{({l,L})}}{z_{i}\left( {l,L} \right)}^{2}} + {\sum\limits_{i = 1}^{N{({l,R})}}{z_{i}\left( {l,R} \right)}^{2}}} & {{\sum\limits_{i = 1}^{N{({l,L})}}{z_{i}\left( {l,L} \right)}} + {\sum\limits_{i = 1}^{N{({l,R})}}{z_{i}\left( {l,R} \right)}}} \\{{\sum\limits_{i = 1}^{N{({l,L})}}{z_{i}\left( {l,L} \right)}} + {\sum\limits_{i = 1}^{N{({l,R})}}{z_{i}\left( {l,R} \right)}}} & {{N\left( {l,L} \right)} + {N\left( {l,R} \right)}}\end{pmatrix}$

The result is quite simple, a matrix containing the accumulated sensorpositions, the accumulated sensor positions squared, and the number ofmeasurements for laser l. For this case, separate L and R sensor sumsare not required.

${{{\mathbb{Z}}^{T}(l)}{{\mathbb{Z}}(l)}} = \begin{pmatrix}{\sum\limits_{i}\left( z_{i} \right)^{2}} & {\sum\limits_{i}z_{i}} \\{\sum\limits_{i}z_{i}} & {N(l)}\end{pmatrix}$

Second, the Z^(T)(l)i(l) vector for laser-1.

${{Z^{T}(l)}{s(l)}} = \begin{pmatrix}{{\sum\limits_{i = 1}^{N{({l,L})}}{z_{i}\left( {l,L} \right)}} - {\sum\limits_{j = 1}^{N{({l,R})}}{z_{j}\left( {l,R} \right)}}} \\{{N\left( {l,L} \right)} - {N\left( {l,R} \right)}}\end{pmatrix}$

This is also simple, containing the difference between left and rightpositions and the difference between the number of left and right dataitems. Separate L and R sensor sums are required here.

Third, the vector Z^(T)(l)h(l).

This vector contains the correlation between heights and stagecoordinates and the accumulated sum of sensor heights.

${{Z^{T}(l)}{h(l)}} = \begin{pmatrix}{{\sum\limits_{i = 1}^{N{({l,L})}}{{h_{i}\left( {l,L} \right)}{z_{i}\left( {l,L} \right)}}} + {\sum\limits_{j = 1}^{N{({l,R})}}{{h_{j}\left( {l,R} \right)}{z_{i}\left( {l,R} \right)}}}} \\{{\sum\limits_{i = 1}^{N{({l,L})}}{h_{i}\left( {l,L} \right)}} + {\sum\limits_{j = 1}^{N{({l,R})}}{h_{j}\left( {l,R} \right)}}}\end{pmatrix}$

Fourth, the value Σ_(l=1) ⁴s^(T)(l)h(l).

This is the accumulated height difference between L and R sensors.

Σ_(l=1) ⁴ s ^(T)(l)h(l)=Σ_(l=1) ⁴{Σ_(i=1) ^(N)(l,L)h _(i)(l, L)−Σ_(i=1)^(N)(l,R)h _(i)(l, R)}

Finally, the value Σ_(l=1) ⁴s^(T)(l)s(l).

This is the difference between the number of left and right data points.

Σ_(l=1) ⁴ s ^(T)(l)s(l)=Σ_(l=1) ⁴ {N(l,R)+N(l,L)}=N(L)+N(R)=N

Implementation—Unique Accumulated Values

A look back at the required terms shows there are only five differentterms that each need to be accumulated for each set of laser and sensorindices. This makes a total of 40 unique values.

${{sSqZ}\left( {l,s} \right)} = {{\sum\limits_{i = 1}^{N{({l,s})}}{\left( {z_{i}\left( {l,s} \right)} \right)^{2}.{{sZ}\left( {l,s} \right)}}} = {{\sum\limits_{i = 1}^{N{({l,s})}}{{z_{i}\left( {l,s} \right)}.{N\left( {l,s} \right)}}} = {\sum\limits_{i = 1}^{N{({l,s})}}1.}}}$${{sHZ}\left( {l,s} \right)} = {{\sum\limits_{i = 1}^{N{({l,s})}}{{h_{i,}\left( {l,s} \right)}{{z_{i}\left( {l,s} \right)}.{{sH}\left( {l,s} \right)}}}} = {\sum\limits_{i = 1}^{N{({l,s})}}{{h_{i}\left( {l,s} \right)}.}}}$

Sums and differences of terms can be expressed in a simple notation.

If the sensor argument is suppressed, then the sum of L and R sensors isindicated as:

sSqZ(l)=sSqZ(l, L)+sSqZ(l, R).

If the laser argument is suppressed, then the sum over all lasers isindicated as:

sH(L)−sH(R)=Σ_(l=1) ⁴ {sH(l, L)−sH(l, R)}.

The symbol Δ indicates a left sensor minus right sensor difference as:

Δ N(l) = N(l, L) − N(l, R), Δ N = N(L) − N(R), Δ sZ(l) = sZ(l, L) − sZ(l, R).

With these accumulated values

${{{Z^{T}(l)}{Z(l)}} = \begin{pmatrix}{{sSqZ}(l)} & {{sZ}(l)} \\{{sZ}(l)} & {N(l)}\end{pmatrix}},\begin{matrix}{{{Z^{T}(l)}{S(l)}} = \begin{pmatrix}{{{sZ}\left( {l,Z} \right)} - {{sZ}\left( {l,R} \right)}} \\{{N\left( {l,L} \right)} - {N\left( {l,R} \right)}}\end{pmatrix}} \\{{= \begin{pmatrix}{\Delta \; {{sZ}(l)}} \\{\Delta \; {N(l)}}\end{pmatrix}},}\end{matrix}$ ${{{Z^{T}(l)}{h(l)}} = \begin{pmatrix}{{sHZ}(l)} \\{{sH}(l)}\end{pmatrix}},{\begin{matrix}{{\sum\limits_{l = 1}^{4}{{s^{T}(l)}{h(l)}}} = {{{sH}(L)} - {{sH}(R)}}} \\{{= {\Delta \; {sH}}},}\end{matrix}\begin{matrix}{{\sum\limits_{l = 1}^{4}{{s^{T}(l)}{s(l)}}} = {{N(L)} + {N(R)}}} \\{= {N.}}\end{matrix}}$

Projection of 3-D coordinates to Laser Coordinates

The parameters a(i), b(i) specified as eight parameters of the above9-parameter fit are actually projections of the 3-D thread axisparameters a_(x), a_(y), b_(x), b_(y). A projection matrix, P, definesthe mapping from the 5-parameter fit to the 9-parameter fit. The anglesa_(i) are the angles of the laser beams with respect to the stage (x,y)axes. For example, laser-1 is incident at 22.5 degrees, laser-2 at 67.5degrees, laser-3 at 112.5 degrees, and laser-4 at 157.4 degrees.

$\begin{matrix}{\begin{pmatrix}{a(1)} \\{b(1)} \\{a(2)} \\{b(2)} \\{a(3)} \\{b(3)} \\{a(4)} \\{b(4)} \\r\end{pmatrix} = {P\begin{pmatrix}a_{x} \\a_{y} \\b_{x} \\b_{y} \\r\end{pmatrix}}} \\{= {\begin{pmatrix}{- {\sin \left( \alpha_{1} \right)}} & {\cos \left( \alpha_{1} \right)} & 0 & 0 & 0 \\0 & 0 & {- {\sin \left( \alpha_{1} \right)}} & {\cos \left( \alpha_{1} \right)} & 0 \\{- {\sin \left( \alpha_{2} \right)}} & {\cos \left( \alpha_{2} \right)} & 0 & 0 & 0 \\0 & 0 & {- {\sin \left( \alpha_{2} \right)}} & {\cos \left( \alpha_{2} \right)} & 0 \\{- {\sin \left( \alpha_{3} \right)}} & {\cos \left( \alpha_{3} \right)} & 0 & 0 & 0 \\0 & 0 & {- {\sin \left( \alpha_{3} \right)}} & {\cos \left( \alpha_{3} \right)} & 0 \\{- {\sin \left( \alpha_{4} \right)}} & {\cos \left( \alpha_{4} \right)} & 0 & 0 & 0 \\0 & 0 & {- {\sin \left( \alpha_{4} \right)}} & {\cos \left( \alpha_{4} \right)} & 0 \\0 & 0 & 0 & 0 & 1\end{pmatrix}\begin{pmatrix}a_{x} \\a_{y} \\b_{x} \\b_{y} \\r\end{pmatrix}}}\end{matrix}$

Using the projection matrix we can reduce the previous matrix vectorequation from a 9-dimension problem to a 5-dimension problem.

$\begin{matrix}{{Z\begin{pmatrix}{a(1)} \\{b(1)} \\{a(2)} \\{b(2)} \\{a(3)} \\{b(3)} \\{a(4)} \\{b(4)} \\r\end{pmatrix}} = {Z\; {P\begin{pmatrix}a_{x} \\a_{y} \\b_{x} \\b_{y} \\r\end{pmatrix}}}} \\{= H}\end{matrix}$

For just one laser, the equation is quite simple.

z_(i)(l, s)a(l) + b(l) + δ(s)r = h_(i)(l, s), wherel ≡ laser  index, s ≡ sensor  index, i ≡ data  index, a(l) = −sin (α_(l))a_(x) + cos (α_(l))a_(y,)b(l) = −sin (α_(l))b_(x) + cos (α l)b_(y), and $\begin{matrix}{{\delta (s)} = {\left( {+ 1} \right)\mspace{14mu} {for}\mspace{14mu} \left( {s=={Left}} \right)}} \\{= {\left( {- 1} \right)\mspace{14mu} {for}\mspace{14mu} {\left( {s=={Right}} \right).}}}\end{matrix}$

This is the least squares solution:

(a _(x) a _(y) b _(x) b _(y) r)^(T)=(P ^(T)(Z ^(T) Z)P)⁻¹(P ^(T) Z^(T))H.

The vector (P^(T)Z^(T))H can be computed in the simplified valuenotation. It is just the linear combination of sumHZ(l) or sumH(l) termsweighted by cos(α_(l)) or sin(α_(l)) factors.

${P^{T}\left( {Z^{T}H} \right)} = {\begin{pmatrix}{- {\sin \left( \alpha_{1} \right)}} & 0 & {- {\sin \left( \alpha_{2} \right)}} & 0 & {- {\sin \left( \alpha_{3} \right)}} & 0 & {- {\sin \left( \alpha_{4} \right)}} & 0 & 0 \\{\cos \left( \alpha_{1} \right)} & 0 & {\cos \left( \alpha_{2} \right)} & 0 & {\cos \left( \alpha_{3} \right)} & 0 & {\cos \left( \alpha_{4} \right)} & 0 & 0 \\0 & {- {\sin \left( \alpha_{1} \right)}} & 0 & {- {\sin \left( \alpha_{2} \right)}} & 0 & {- {\sin \left( \alpha_{3} \right)}} & 0 & {- {\sin \left( \alpha_{4} \right)}} & 0 \\0 & {\cos \left( \alpha_{1} \right)} & 0 & {\cos \left( \alpha_{2} \right)} & 0 & {\cos \left( \alpha_{3} \right)} & 0 & {\cos \left( \alpha_{4} \right)} & 0 \\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{pmatrix}\begin{pmatrix}{{sHZ}(1)} \\{{sH}(1)} \\{{sHZ}(2)} \\{{sH}(2)} \\{{sHZ}(3)} \\{{sH}(3)} \\{{sHZ}(4)} \\{{sH}(4)} \\{\Delta \; {sH}}\end{pmatrix}}$ ${P^{T}\left( {Z^{T}H} \right)} = \begin{pmatrix}{- {\sum\limits_{l = 1}^{4}{{\sin \left( \alpha_{l} \right)}{{sHZ}(l)}}}} \\{+ {\sum\limits_{l = 1}^{4}{{\cos \left( \alpha_{l} \right)}{{sHZ}(l)}}}} \\{- {\sum\limits_{l = 1}^{4}{{\sin \left( \alpha_{l} \right)}{{sH}(l)}}}} \\{+ {\sum\limits_{l = 1}^{4}{{\cos \left( \alpha_{l} \right)}{{sH}(l)}}}} \\{\Delta \; {sH}}\end{pmatrix}$

The matrix (P^(T)Z^(T)ZP) also can be computed in the simplified valuenotation.

$\left( {{P^{T}\left( {Z^{T}Z} \right)}P} \right) = {{P^{T}\begin{pmatrix}{{sSqZ}(1)} & {{sZ}(1)} & 0 & 0 & 0 & 0 & 0 & 0 & {\Delta \; {{sZ}(1)}} \\{{sZ}(1)} & {N(1)} & 0 & 0 & 0 & 0 & 0 & 0 & {\Delta \; {N(1)}} \\0 & 0 & {{sSqZ}(2)} & {{sZ}(2)} & 0 & 0 & 0 & 0 & {\Delta \; {{sZ}(2)}} \\0 & 0 & {{sZ}(2)} & {N(2)} & 0 & 0 & 0 & 0 & {\Delta \; {N(2)}} \\0 & 0 & 0 & 0 & {{sSqZ}(3)} & {{sZ}(3)} & 0 & 0 & {\Delta \; {{sZ}(3)}} \\0 & 0 & 0 & 0 & {{sZ}(3)} & {N(3)} & 0 & 0 & {\Delta \; {N(3)}} \\0 & 0 & 0 & 0 & 0 & 0 & {{sSqZ}(4)} & {{sZ}(4)} & {\Delta \; {{sZ}(4)}} \\0 & 0 & 0 & 0 & 0 & 0 & {{sZ}(4)} & {N(4)} & {\Delta \; {N(4)}} \\{\Delta \; {{sZ}(1)}} & {\Delta \; {N(1)}} & {\Delta \; {{sZ}(2)}} & {\Delta \; {N(2)}} & {\Delta \; {{sZ}(3)}} & {\Delta \; {{sZ}(4)}} & {\Delta \; {{sZ}(4)}} & {\Delta \; ({N4})} & N\end{pmatrix}}\begin{pmatrix}{- {\sin \left( \alpha_{1} \right)}} & {\cos \left( \alpha_{1} \right)} & 0 & 0 & 0 \\0 & 0 & {- {\sin \left( \alpha_{1} \right)}} & {\cos \left( \alpha_{1} \right)} & 0 \\{- {\sin \left( \alpha_{2} \right)}} & {\cos \left( \alpha_{2} \right)} & 0 & 0 & 0 \\0 & 0 & {- {\sin \left( \alpha_{2} \right)}} & {\cos \left( \alpha_{2} \right)} & 0 \\{- {\sin \left( \alpha_{3} \right)}} & {\cos \left( \alpha_{3} \right)} & 0 & 0 & 0 \\0 & 0 & {- {\sin \left( \alpha_{3} \right)}} & {\cos \left( \alpha_{3} \right)} & 0 \\{- {\sin \left( \alpha_{4} \right)}} & {\cos \left( \alpha_{4} \right)} & 0 & 0 & 0 \\0 & 0 & {- {\sin \left( \alpha_{4} \right)}} & {\cos \left( \alpha_{4} \right)} & 0 \\0 & 0 & 0 & 0 & 1\end{pmatrix}}$

intermediate result:

$\left. {\left( {{P^{T}\left( {Z^{T}Z} \right)}P} \right) = P^{T}} \right)\begin{pmatrix}{{- {\sin \left( \alpha_{1} \right)}}{{sSqZ}(1)}} & {{\cos \left( \alpha_{1} \right)}{{sSqZ}(1)}} & {{- {\sin \left( \alpha_{1} \right)}}{{sZ}(1)}} & {{\cos \left( \alpha_{1} \right)}{{sZ}(1)}} & {\Delta \; {{sZ}(1)}} \\{{- {\sin \left( \alpha_{1} \right)}}{{sZ}(1)}} & {{\cos \left( \alpha_{1} \right)}{{sZ}(1)}} & {{- {\sin \left( \alpha_{1} \right)}}{N(1)}} & {{\cos \left( \alpha_{1} \right)}{N(1)}} & {\Delta \; {N(1)}} \\{{- {\sin \left( \alpha_{2} \right)}}{{sSqZ}(2)}} & {{\cos \left( \alpha_{2} \right)}{{sSqZ}(2)}} & {{- {\sin \left( \alpha_{2} \right)}}{{sZ}(2)}} & {{\cos \left( \alpha_{2} \right)}{{sZ}(2)}} & {\Delta \; {{sZ}(2)}} \\{{- {\sin \left( \alpha_{2} \right)}}{{sZ}(2)}} & {{\cos \left( \alpha_{2} \right)}{{sZ}(2)}} & {{- {\sin \left( \alpha_{2} \right)}}{N(2)}} & {{\cos \left( \alpha_{2} \right)}{N(2)}} & {\Delta \; {N(2)}} \\{{- {\sin \left( \alpha_{3} \right)}}{{sSqZ}(3)}} & {{\cos \left( \alpha_{3} \right)}{{sSqZ}(3)}} & {{- {\sin \left( \alpha_{3} \right)}}{{sZ}(3)}} & {{\cos \left( \alpha_{3} \right)}{{sZ}(3)}} & {\Delta \; {{sZ}(3)}} \\{{- {\sin \left( \alpha_{3} \right)}}{{sZ}(3)}} & {{\cos \left( \alpha_{3} \right)}{{sZ}(3)}} & {{- {\sin \left( \alpha_{3} \right)}}{{sN}(3)}} & {{\cos \left( \alpha_{3} \right)}{N(3)}} & {\Delta \; {N(3)}} \\{{- {\sin \left( \alpha_{4} \right)}}{{sSqZ}(4)}} & {{\cos \left( \alpha_{4} \right)}{{sSqZ}(4)}} & {{- {\sin \left( \alpha_{4} \right)}}{{sZ}(4)}} & {{\cos \left( \alpha_{4} \right)}{{sZ}(4)}} & {\Delta \; {{sZ}(4)}} \\{{- {\sin \left( \alpha_{4} \right)}}{{sZ}(4)}} & {{\cos \left( \alpha_{4} \right)}{{sZ}(4)}} & {{- {\sin \left( \alpha_{4} \right)}}{N(4)}} & {{\cos \left( \alpha_{4} \right)}{N(4)}} & {\Delta \; {N(4)}} \\{- {\sum\limits_{l = 1}^{4}{{\sin \left( \alpha_{1} \right)}\Delta \; {{sZ}(l)}}}} & {- {\sum\limits_{l = 1}^{4}{{\cos \left( \alpha_{1} \right)}\Delta \; {{sZ}(l)}}}} & {- {\sum\limits_{l = 1}^{4}{{\sin \left( \alpha_{1} \right)}\Delta \; {N(l)}}}} & {- {\sum\limits_{l = 1}^{4}{{\cos \left( \alpha_{1} \right)}\Delta \; {N(l)}}}} & N\end{pmatrix}$

The final result is a 5×5 matrix with the sin( ) and cos( ) terms mixingthe data from individual lasers.

$\left( {{P^{T}\left( {Z^{T}Z} \right)}P} \right) = \left( \begin{matrix}{+ {\sum\limits_{l = 1}^{4}{{\sin^{2}\left( \alpha_{1} \right)}{{sSqZ}(l)}}}} & {- {\sum\limits_{l = 1}^{4}{{\sin \left( \alpha_{1} \right)}{\cos \left( \alpha_{1} \right)}{{sSqZ}(l)}}}} & {+ {\sum\limits_{l = 1}^{4}{{\sin^{2}\left( \alpha_{1} \right)}{{sZ}(l)}}}} & {- {\sum\limits_{l = 1}^{4}{{\sin \left( \alpha_{1} \right)}{\cos \left( \alpha_{1} \right)}{{sZ}(l)}}}} & {+ {\sum\limits_{l = 1}^{4}{{\sin \left( \alpha_{1} \right)}\Delta \; {{sZ}(l)}}}} \\{- {\sum\limits_{l = 1}^{4}{{\sin \left( \alpha_{1} \right)}{\cos \left( \alpha_{1} \right)}{{sSqZ}(l)}}}} & {+ {\sum\limits_{l = 1}^{4}{{\cos^{2}\left( \alpha_{1} \right)}{{sSqZ}(l)}}}} & {- {\sum\limits_{l = 1}^{4}{{\sin \left( \alpha_{1} \right)}{\cos \left( \alpha_{1} \right)}{{sZ}(l)}}}} & {+ {\sum\limits_{l = 1}^{4}{{\cos^{2}\left( \alpha_{1} \right)}{{sZ}(l)}}}} & {+ {\sum\limits_{l = 1}^{4}{{\cos \left( \alpha_{1} \right)}\Delta \; {{sZ}(l)}}}} \\{+ {\sum\limits_{l = 1}^{4}{{\sin^{2}\left( \alpha_{1} \right)}{{sZ}(l)}}}} & {- {\sum\limits_{l = 1}^{4}{{\sin \left( \alpha_{1} \right)}{\cos \left( \alpha_{1} \right)}{{sZ}(l)}}}} & {+ {\sum\limits_{l = 1}^{4}{{\sin^{2}\left( \alpha_{1} \right)}{N(l)}}}} & {- {\sum\limits_{l = 1}^{4}{{\sin \left( \alpha_{1} \right)}{\cos \left( \alpha_{1} \right)}{N(l)}}}} & {- {\sum\limits_{l = 1}^{4}{{\sin \left( \alpha_{1} \right)}\Delta \; {N(l)}}}} \\{- {\sum\limits_{l = 1}^{4}{{\sin \left( \alpha_{1} \right)}{\cos \left( \alpha_{1} \right)}{{sZ}(l)}}}} & {+ {\sum\limits_{l = 1}^{4}{{\cos^{2}\left( \alpha_{1} \right)}{{sZ}(l)}}}} & {- {\sum\limits_{l = 1}^{4}{{\sin \left( \alpha_{1} \right)}{\cos \left( \alpha_{1} \right)}{N(l)}}}} & {+ {\sum\limits_{l = 1}^{4}{{\cos^{2}\left( \alpha_{1} \right)}{N(l)}}}} & {+ {\sum\limits_{l = 1}^{4}{{\cos \left( \alpha_{1} \right)}\Delta \; {N(l)}}}} \\{- {\sum\limits_{l = 1}^{4}{{\sin \left( \alpha_{1} \right)}\Delta \; {{sZ}(l)}}}} & {+ {\sum\limits_{l = 1}^{4}{{\cos \left( \alpha_{1} \right)}\Delta \; {{sZ}(l)}}}} & {- {\sum\limits_{l = 1}^{4}{{\sin \left( \alpha_{1} \right)}\Delta \; {N(l)}}}} & {+ {\sum\limits_{l = 1}^{4}{{\cos \left( \alpha_{1} \right)}\Delta \; {N(l)}}}} & N\end{matrix} \right)$

One can define a set of 4-element vectors to make the previous resultslook more compact.

$\sin\limits^{\rightharpoonup} = \left( {{\sin \left( \alpha_{1} \right)}\mspace{14mu} {\sin \left( \alpha_{2} \right)}\mspace{14mu} {\sin \left( \alpha_{3} \right)}\mspace{14mu} {\sin \left( \alpha_{4} \right)}} \right)$$\cos\limits^{\rightharpoonup} = \left( {{\cos \left( \alpha_{1} \right)}\mspace{14mu} {\cos \left( \alpha_{2} \right)}\mspace{14mu} {\cos \left( \alpha_{3} \right)}\mspace{14mu} {\cos \left( \alpha_{4} \right)}} \right)$$\overset{\rightharpoonup}{\left( {\sin \; \cos} \right)} = \begin{pmatrix}{\sin \left( \alpha_{1} \right){\cos \left( \alpha_{1} \right)}\mspace{14mu} {\sin \left( \alpha_{2} \right)}{\cos \left( \alpha_{2} \right)}} \\{\sin \left( \alpha_{3} \right){\cos \left( \alpha_{3} \right)}\mspace{14mu} {\sin \left( \alpha_{4} \right)}{\cos \left( \alpha_{4} \right)}}\end{pmatrix}$$\overset{\rightharpoonup}{\left( \sin \right)^{2}} = \left( {{\sin^{2}\left( \alpha_{1} \right)}\mspace{14mu} {\sin^{2}\left( \alpha_{2} \right)}\mspace{14mu} {\sin^{2}\left( \alpha_{3} \right)}\mspace{14mu} {\sin^{2}\left( \alpha_{4} \right)}} \right)$$\overset{\rightharpoonup}{\left( \; \cos \right)^{2}} = \left( {{\cos^{2}\left( \alpha_{1} \right)}\mspace{14mu} {\cos^{2}\left( \alpha_{2} \right)}\mspace{14mu} {\cos^{2}\left( \alpha_{3} \right)}\mspace{14mu} {\cos^{2}\left( \alpha_{4} \right)}} \right)$$\overset{\rightharpoonup}{sSqZ} = \left( {{{sSqZ}(1)}\mspace{14mu} {{sSqZ}(2)}\mspace{14mu} {{sSqZ}(3)}\mspace{14mu} {{sSqZ}(4)}} \right)$$\overset{\rightharpoonup}{sZ} = \left( {{{sZ}(1)}\mspace{14mu} {{sZ}(2)}\mspace{14mu} {{sZ}(3)}\mspace{14mu} {{sZ}(4)}} \right)$$\overset{\rightharpoonup}{sH} = \left( {{{{sH}(1)}\mspace{14mu} {{sH}(2)}\mspace{14mu} {{sH}(3)}\mspace{14mu} {{sH}(4)}\overset{\rightharpoonup}{sHZ}} = {{\left( {{{sHZ}(1)}\mspace{14mu} {{sHZ}(2)}\mspace{14mu} {{sHZ}(3)}\mspace{14mu} {{sHZ}(4)}} \right)\overset{\rightharpoonup}{N}} = {{\left( {{N(1)}\mspace{14mu} {N(2)}\mspace{14mu} {N(3)}\mspace{14mu} {N(4)}} \right)\overset{\rightharpoonup}{\Delta \; N}} = {{\left( {\Delta \; {N(1)}\mspace{14mu} \Delta \; {N(2)}\mspace{14mu} \Delta \; {N(3)}\mspace{14mu} \Delta \; {N(4)}} \right)\Delta \; {sH}} = {{\left( {\Delta \; {{sH}(1)}\mspace{14mu} \Delta \; {{sH}(2)}\mspace{14mu} \Delta \; {{sH}(3)}\mspace{14mu} \Delta \; {{sH}(4)}} \right)\Delta \; {sZ}} = \left( {\Delta \; {{sZ}(1)}\mspace{14mu} \Delta \; {{sZ}(2)}\mspace{14mu} \Delta \; {{sZ}(3)}\mspace{14mu} \Delta \; {{sZ}(4)}} \right)}}}}} \right.$

The final results are below.

Every term is the dot product of a geometrical vector representing theincident angles of each of the four lasers and a data vectorrepresenting what is observed in each of the four lasers. There arefifteen independent numbers to be calculated for (P^(T)(Z^(T)Z)P) andfive for P^(T)(Z^(T)H).

${P^{T}\left( {Z^{T}H} \right)} = \begin{pmatrix}{{- \sin\limits^{\rightharpoonup}} \cdot \overset{\rightharpoonup}{sHZ}} \\{{+ \cos\limits^{\rightharpoonup}} \cdot \overset{\rightharpoonup}{sHZ}} \\{{- \sin\limits^{\rightharpoonup}} \cdot \overset{\rightharpoonup}{sH}} \\{{+ \cos\limits^{\rightharpoonup}} \cdot \overset{\rightharpoonup}{sH}} \\{{+ \Delta}\; {sH}}\end{pmatrix}$$\left( {{P^{T}\left( {Z^{T}Z} \right)}P} \right) = \left( \begin{matrix}{{+ \overset{\rightharpoonup}{\left( \sin \right)^{2}}} \cdot \overset{\rightharpoonup}{sSqZ}} & {{- \overset{\rightharpoonup}{\left( {\sin \; \cos} \right)}} \cdot \overset{\rightharpoonup}{sSqZ}} & {{+ \overset{\rightharpoonup}{\left( \sin \right)^{2}}} \cdot \overset{\rightharpoonup}{sZ}} & {{- \overset{\rightharpoonup}{\left( {\sin \; \cos} \right)}} \cdot \overset{\rightharpoonup}{sZ}} & {{- \sin\limits^{\rightharpoonup}} \cdot \overset{\rightharpoonup}{\Delta \; {sZ}}} \\{{- \overset{\rightharpoonup}{\left( {\sin \; \cos} \right)}} \cdot \overset{\rightharpoonup}{sSqZ}} & {{+ \overset{\rightharpoonup}{\left( \cos \right)^{2}}} \cdot \overset{\rightharpoonup}{sSqZ}} & {{- \overset{\rightharpoonup}{\left( {\sin \; \cos} \right)}} \cdot \overset{\rightharpoonup}{sZ}} & {{+ \overset{\rightharpoonup}{\left( \; \cos \right)^{2}}} \cdot \overset{\rightharpoonup}{sZ}} & {{+ \cos\limits^{\rightharpoonup}} \cdot \overset{\rightharpoonup}{\Delta \; {sZ}}} \\{{+ \overset{\rightharpoonup}{\left( \sin \right)^{2}}} \cdot \overset{\rightharpoonup}{sZ}} & {{- \overset{\rightharpoonup}{\left( {\sin \; \cos} \right)}} \cdot \overset{\rightharpoonup}{sZ}} & {{+ \overset{\rightharpoonup}{\left( \sin \right)^{2}}} \cdot \overset{\rightharpoonup}{N}} & {{- \overset{\rightharpoonup}{\left( {\sin \; \cos} \right)}} \cdot \overset{\rightharpoonup}{N}} & {{- \sin\limits^{\rightharpoonup}} \cdot \overset{\rightharpoonup}{\Delta \; N}} \\{{- \overset{\rightharpoonup}{\left( {\sin \; \cos} \right)}} \cdot \overset{\rightharpoonup}{sZ}} & {{+ \overset{\rightharpoonup}{\left( \cos \right)^{2}}} \cdot \overset{\rightharpoonup}{sZ}} & {{- \overset{\rightharpoonup}{\left( {\sin \; \cos} \right)}} \cdot \overset{\rightharpoonup}{N}} & {{+ \overset{\rightharpoonup}{\left( \; \cos \right)^{2}}} \cdot \overset{\rightharpoonup}{N}} & {{+ \cos\limits^{\rightharpoonup}} \cdot \overset{\rightharpoonup}{\Delta \; N}} \\{{- \sin\limits^{\rightharpoonup}} \cdot \overset{\rightharpoonup}{\Delta \; {sZ}}} & {{+ \cos\limits^{\rightharpoonup}} \cdot \overset{\rightharpoonup}{\Delta \; {sZ}}} & {{- \sin\limits^{\rightharpoonup}} \cdot \overset{\rightharpoonup}{\Delta \; N}} & {{+ \cos\limits^{\rightharpoonup}} \cdot \overset{\rightharpoonup}{\Delta \; N}} & {+ N}\end{matrix} \right)$

Simple Cases

Suppose the cylinder is exactly aligned with the z-axis, and the data isexactly centered above and below each sensor's center line. Then in eachsensor, the left sensor will have a measurement points of type (x, h)and the right sensor will have measurements of type (x,−h). This willmean that the sHZ and sH terms will be zero. The term ΔsH will be a sumof +h and -(−h) terms or ΔsH≈Nh.

The solution equation would then read:

$\begin{matrix}{{\left( {{P^{T}\left( {Z^{T}Z} \right)}P} \right)\begin{pmatrix}a_{x} \\a_{y} \\b_{x} \\b_{y} \\r\end{pmatrix}} = {\begin{pmatrix}\ldots & \ldots & \ldots & \ldots & 0 \\\ldots & \ldots & \ldots & \ldots & 0 \\\ldots & \ldots & \ldots & \ldots & 0 \\\ldots & \ldots & \ldots & \ldots & 0 \\0 & 0 & 0 & 0 & N\end{pmatrix}\begin{pmatrix}a_{x} \\a_{y} \\b_{x} \\b_{y} \\r\end{pmatrix}}} \\{= {P\left( {Z^{T}H} \right)}} \\{= \begin{pmatrix}{{- \sin\limits^{\rightharpoonup}} \cdot \overset{\rightharpoonup}{sHZ}} \\{{+ \cos\limits^{\rightharpoonup}} \cdot \overset{\rightharpoonup}{sHZ}} \\{{- \sin\limits^{\rightharpoonup}} \cdot \overset{\rightharpoonup}{sH}} \\{{+ \cos\limits^{\rightharpoonup}} \cdot \overset{\rightharpoonup}{sH}} \\{{+ \Delta}\; {sH}}\end{pmatrix}} \\{= \begin{pmatrix}0 \\0 \\0 \\0 \\{Nh}\end{pmatrix}}\end{matrix}$

This has the solution

a_(x)=a_(y)=b_(x)=b_(y)=0,

r=Nh/N=h.

APPENDIX D

The method of assembly and alignment of the optical and mechanicalcomponents of the light plane generator receiver and light planegenerator modules is executed utilizing the alignment fixture 100 ofFIG. 31. The alignment fixture includes an optical rail, modifiedbreadboards 120, 151 and 180, and a plurality of custom and commerciallyavailable opto-mechanical positioning devices. A detector assemblygenerally indicated at 159 and located on breadboard 151 replicates thefunction of a light plane receiver module.

The alignment fixture 100 further includes a references laser 118,aligned such that the center line of its light beam is horizontallyparallel to the reference breadboards 120 and 151 and lens and mirrormount 304 interfaces. Preferably, the laser 118 is a spatially filteredsolid state laser such as the 40001 available from LumenFlow Corp. ofMiddleville, Mich.

Also shown supported on the breadboard 120 is a reference prism (i.e.,rhomboid prism) assembly 122, the use of which is detailed in thisapplication. As it is the method of assembly that is important to properfunctioning modules, this disclosure emphasizes the interfacingcomponents of the fixture while focusing on the detailed steps of themethod to produce the modules.

Alignment fixture 100 of FIG. 31 includes a rail and a plurality ofstage assemblies, generally indicated at 102, which, in turn, includes arail assembly, generally indicated at 104. The rail assembly 104includes a rail 106 which is supported by pairs of bench legs 108. Thealignment fixture 100 also includes a set of rail carriages 110 slidablymounted on the rail 106.

The alignment fixture 100 further includes a transmitter stage assembly,generally indicated at 112. The transmitter stage assembly 112 includesan L1 (i.e., first lens) manipulator stage assembly, generally indicatedat 113, an L2 (i.e., second lens) manipulator stage assembly 114 and anL3 (i.e., third lens) manipulator stage assembly 116.

Each of the assemblies 113, 114 and 116 are supported together with thelaser 118 and its support bracket 119 on the breadboard or substrate120.

Also shown supported on the breadboard 120 is the reference prism (i.e.,rhomboid prism) assembly 122, an alignment aperture 124, a filter mount126 and a post holder 128.

The precision rhomboid prism assembly 122 is important to the assemblyof the modules, calibration of the align and focus instrument, and toassembly of the optical head.

The assembly 122 includes two metal plates, two dowel pins, and arhomboid prism.

The rhomboid prism assembly 122 is the gage that establishes the heightdifference between the height of laser light entering the transmitter(through lens 316) (preferably, 0.984″) and the height of laser lightexiting the transmitter (through lens 310) (preferably, 1.679″).

The height is set by precisely machined dowel pin positions and theprecise shape of the rhomboid.

The rhomboid prism has the property that a straight beam of lightentering the prism exists the prism in a straight line and in a pathexactly parallel to the beam path of the entry beam. The heightdifference between the entry and exit beams is set by the rotationalangle of the prism, relative to the entry beam.

The alignment fixture 100 also includes a clamp post assembly 130 with akinematic ball also supported on the breadboard 120.

The alignment fixture 100 further includes a relay telescope assembly,generally indicated at 132, which, in turn, includes a relay doubletassembly 134 and an IR doublet 136. The relay telescope 132 is mountedon its carriage 110 by a scope mount 138.

Mounted on the rail 106 is another carriage 110 on which a post 140 issupported at one end of the fixture 100. The post 140 supports a filteror target holder 142. In turn, the holder 142 supports a target 210.

The alignment fixture 100 also includes a receiver stage assembly 150which, in turn, includes the breadboard or substrate 151. The receiverstage assembly 150 includes an alignment assembly 152 including analignment aperture 154.

The receiver stage assembly 150 further includes an optical rail 156supported on the breadboard 151. The detector assembly, generallyindicated at 159, is adjustably mounted on the rail 156. The detectorassembly 159 includes electronic boards 157, together with the sensormount with an aperture 155. The receiver stage assembly 150 furtherincludes a filter mount 160 and a vertical slit 162.

The receiver stage assembly 150 also includes a manipulator bracket 164and an assembly 166 having a kinematic ball 168 mounted at a distal endthereof.

The receiver stage assembly 150 further includes an L4 (i.e., fourthlens) manipulator stage assembly 170.

The L1 manipulator stage assembly 113 includes the breadboard 180 onwhich an x-y stage assembly 184, together with a kinematic base 186 aremounted. Adjustment screws 182 are provided to adjust the position ofthe stage assembly 184. Clamping arms 188 are mounted at a distal end ofthe stage assembly 113.

FIGS. 31 and 37 taken together illustrate: a clamp for lens 310 attachedto screw 116; a screw attached to clamp 114 below plate 120; arotational screw to rod attached to clamp 188; lower two screw holes of322 establish a rotational adjustment axis for 322; and upper one screwhole of 322 establishes control of rotation about 322 adjustment axis.

Overview of the Transmitter Alignment Process

The alignment of the transmitter module's optical components isimportant to the operation of the Laser Lab system. The alignment isaccomplished with the align and focus (i.e., A&F) instrument or fixture100.

The transmitter module is mounted in the A&F instrument 100, usingreference surfaces “A”, “B”, and “C” of the module. Reference surface“A” mounts flat to base plate 120, which has been aligned parallel tothe laser beam used in the A&F instrument 100, generated by the laser118. Reference surface “C” is made flush to two kinematic mounts thathave been aligned parallel to the laser beam 118. Reference surface “B”is flush to one kinematic mount and establishes the correct positionalong the beam line of the laser 118.

Mirror-2 is mounted to reference surface “E” on module 304. Mirror-1 ismounted to plate 322 which is mounted to reference surface “F”.

In the alignment process the optical components are configured to meetsystem requirements using the process previously detailed. During thealignment process the A&F instrument 100 holds the parts in place with aset of clamps and piece holders. When the alignment process iscompleted, the optical components are fixed to the surfaces of module304 with a glue that permanently holds them in place.

The following is a list of transmitter optical module components andtheir adjustments that are fixed in the align and focus instrument 100:

-   -   first cylindrical lens 316: plate 318 is clamped in A&F part        188. 188 can be moved in three directions: linearly parallel to        laser beam and linearly perpendicular to laser beam (with screws        182), and rotated about the axis of the rod holding clamp 188.    -   second cylindrical lens 312: linearly perpendicular to laser        beam with screw below plate 120, attached to 114. The lens can        also be rotated within clamp 114.    -   third cylindrical lens 310: linearly perpendicular to laser        beam, in up-down relation to plate 120, by screw 116. The lens        can also be rotated within clamp attached to screw 116.    -   adjustable mirror mount 322: radial adjustment about axis        established by lower two screws mounting 322 to 304. Adjustment        controlled by top screw of 322.

Thus, there are a total of eight independent adjustment parameters forthe transmitter module. Each parameter is optimized in the alignmentprocess, and fixed with glue before removal of the module 66 from theA&F instrument 100.

1.0 Produce light plane generator module assembly:

-   -   1.1 Install machined lens and mirror mount 304 to breadboard        120. Place an alignment aperture in the reference laser beam        path so that the laser beam passes through the aperture without        creating distortion.    -   1.2 Install lens 312 within the mount 304 and align so that the        beam passes through the aperture and is coincident with the        laser beam axis. Remove the aperture from the beam path.    -   1.3 Assemble L1 sub-assembly 320 components, cylinder lens 316,        L1 lens mount plate 318 and lens mount base 319. Assemble        sub-assembly 320 to mount 304.    -   1.4 Adjust rotational position of lens 312 about the laser beam        axis so that beam profile is level (FIG. 32).    -   1.5 Adjust rotational position of lens 316 about the one degree        of freedom provided so that the beam profile is flat (FIG. 33).    -   1.6 Adjust position of lens 316 along laser beam axis so that        beam width is visually equal at a distance greater than or equal        to 216″ and at 4″ distance from exit aperture of machined lens        and mirror mount 304.    -   1.7 Adjust the horizontal position of lens 316 so that power        distribution is visually balanced within the beam profile.    -   1.8 Re-adjust the horizontal position of lens 312 so that beam        is centered along the laser beam axis.    -   1.9 Re-adjust position of lens 316 along laser beam axis so that        full angle laser beam divergence is less than 0.25 m radians.

θ=(d2−d1)/(l2−l1)

-   -    where θ is beam divergence in radians; d1 is the horizontal        beam width at distance 1; d2 is the horizontal beam width at        distance 2; l1 is the distance from the output aperture of        machined lens and mirror mount 304 to the beam measurement point        at distance 1; l2 is the distance from the output aperture of        machined lens and mirror mount 304 to the beam measurement point        at distance 2.    -   1.10 Assemble lens 310 to machined lens and mirror mount 304.        Ensure lens 310 is intimate with machined lens and mirror mount        304.    -   1.11 Place a 150 μm slit aperture in laser beam path with slit        parallel to machined lens and mirror mount 304 mounting surface        at a height of 1.679″±0.002″. Adjust vertical position of lens        304 so that beam passes through slit opening. Alignment is        optimum when the maximum amount of light passes through        aperture. Remove slit aperture from beam path.    -   1.12 Adjust position of lens 310 rotationally so that beam        profile is not distorted (i.e., unlike FIG. 35) and generally        forms a rectangle (i.e., like FIG. 36).    -   1.13 Place the detector assembly 159 of the alignment fixture        100 in the laser beam path. Adjust lens 312 so that the        attenuated beam profile is aligned with the laser beam axis.    -   1.14 Adjust lens 316 position horizontally so that output power        is distributed equally ±5 mV across laser beam profile as        measured using detector assembly 159 and dual trace oscilloscope        readout.    -   1.15 Verify collimation, beam height, distortion, power        distribution and beam pointing. Correct as necessary. Apply and        cure adhesives to fixedly mount optical components to the mount        304.    -   1.16 Remove assembled light plane generator module from        alignment fixture 100.

2.0 Produce light plane receiver module assembly:

-   -   2.1 Install reference rhomboid prism 122 to breadboard 120 to        establish laser beam offset height parallel to machined lens and        mirror mount 500 of 1.679″.    -   2.2 Install machined lens and mirror mount 500 to breadboard        151.    -   2.3 Assemble lens 506 to machined lens and mirror mount 500.        Ensure lens 506 is intimate with machined lens and mirror mount        500.    -   2.4 Align lens 506 so that the beam is coincident with the laser        beam axis.    -   2.5 Place a cylinder lens 316 in the beam path utilizing the        cylinder lens holder provided within alignment fixture 100        positioned so that the laser plane output is parallel to the        mounting surface of machined lens and mirror mount 500.    -   2.6 Assemble L5 sub-assembly components, receiver spherical lens        508 (two) and lens mount plate 504 to machined lens and mirror        mount 500 by screws 516 (FIG. 47).    -   2.7 Place a target card with a line scribed at 1.679″±0.002″ at        back surface of machined lens and mirror mount 500.    -   2.8 Adjust lens pair vertically so that laser beam strikes        scribe line.    -   2.9 Adjust horizontal position of lens pair 508 so that laser        beam strikes pair center seam such that resulting spots are        visually equal in brightness. Remove target card.    -   2.10 Assemble aperture elements 510 and Hamamatsu photodetectors        514 to detector mount 502 using screws 505.    -   2.11 Assemble detector mount 502 to machined lens and mirror        mount 500 using screws 505.    -   2.12 Adjust position of detector mount 502 horizontally so that        beams are focused onto detectors 514 without clipping aperture        element 510 edges.    -   2.13 Remove rhomboid prism 122 and cylinder lens 316 from        fixture. Install a selected light plane generator module to        breadboard 120.    -   2.14 Adjust horizontal position of lens pair 508 so that output        power is distributed equally ±5 mV across laser beam profile as        measured using Hamamatsu photodetectors 514 output and dual        trace oscilloscope readout.    -   2.15 Verify that beams are focused onto detectors 514 without        clipping aperture element 510 edges.    -   2.16 Apply and cure adhesives to fixedly mount optical        components to the mount 500.    -   2.17 Remove assembled light plane generator module and light        plane receiver module pair from alignment fixture 100.

While embodiments of the invention have been illustrated and described,it is not intended that these embodiments illustrate and describe allpossible forms of the invention. Rather, the words used in thespecification are words of description rather than limitation, and it isunderstood that various changes may be made without departing from thespirit and scope of the invention.

1. A system for indirectly measuring a geometric dimension related to anopening in an apertured exterior surface of a part based on directmeasurements of the part when fixtured at a measurement station, thesystem comprising: first and second holding devices for holding a parttherebetween in a part-retaining position in which the part is firmlyheld between the devices at the apertured exterior surface wherein aportion of one of the holding devices extends into the opening in thepart-retaining position; a head apparatus including a plurality ofradiation sources for successively directing a first array of planes ofradiation at the one of the holding devices and a second array of planesof radiation at the part in the part-retaining position so that the oneof the holding devices and the part occlude their respective planes ofradiation to create corresponding arrays of unobstructed planar portionsof the planes of radiation wherein each of the unobstructed planarportions contains an amount of radiation which is representative of arespective external geometric dimension of either the one of the holdingdevices or the part, the head apparatus further including a plurality ofreceiver modules for measuring the amount of radiation present in eachof the unobstructed planar portions to obtain holding device and partsignals; a movable stage subsystem coupled to the head apparatus fortranslating the head apparatus relative to the devices and the part sothat the planes of radiation scan the one of the holding devices and thepart in the part-retaining position; a signal processor for processingthe holding device and part signals to obtain data; and a data processorfor processing the data to obtain the direct measurements, the dataprocessor determining the geometric dimension related to the openingbased on the direct measurements.
 2. The system as claimed in claim 1,wherein the holding devices are movable relative to each other betweenthe part-retaining position and a part-release position in which thedevices release the part for removal of the part from the station. 3.The system as claimed in claim 1, wherein the portion of the one of theholding devices has a sloped region at which the portion seats on theopening at the exterior surface and wherein the portion was a crosssectional shape substantially identical to the shape of the opening atthe exterior surface and wherein the sloped region defines a range ofsizes of the opening at the exterior surface that can be indirectlymeasured at the station.
 4. The system as claimed in claim 3, whereinthe portion of the one of the holding devices is frustum-shaped andwherein the geometric dimension is distance across the opening at theexterior surface.
 5. The system as claimed in claim 1, wherein thegeometric dimension related to the opening is thickness of a wall whichdefines the opening at the exterior surface.
 6. The system as claimed inclaim 3, wherein the one of the holding devices includes a spring forspring-loading the portion so that the portion applies a variableclamping force to the part.
 7. The system as claimed in claim 3, whereinthe sloped region has a constant slope.
 8. The system as claimed inclaim 1, further comprising memory coupled to the data processor tostore data which represents height of the portion wherein the firstholding device includes the portion, the portion having a distal endsurface which engages an inner end surface of the opening in thepart-retaining position and wherein the data processor processes thedata including the stored data to obtain depth of the opening in thepart.
 9. The system as claimed in claim 1, wherein the first holdingdevice includes first and second portions which extend into the openingin the part-retaining position, the first portion having a sloped regionand a cross sectional shape substantially identical to the shape of theopening at the exterior surface and wherein the sloped region defines arange of sizes of the opening at the exterior surface that can beindirectly measured at the station and wherein the second portion has adistal end surface which engages an inner end surface of the opening inthe part-retaining position.
 10. The system as claimed in claim 9,wherein the first and second portions are movable relative to each otherand wherein the first holding device further includes a spring forspring loading the first portion so that the first portion can apply avariable clamping force to the part.
 11. The system as claimed in claim1, wherein the part is an ammunition case.
 12. A system for indirectlymeasuring geometric dimensions related to openings in apertured exteriorsurfaces of a part based on direct measurements of the part whenfixtured at a measurement station, the system comprising: first andsecond holding devices for holding a part therebetween in apart-retaining position in which the part is firmly held between thedevices at the apertured exterior surfaces of the part wherein a portionof each of the holding devices extends into its respective opening inthe part-retaining position; a head apparatus including a plurality ofradiation sources for successively directing a first array of planes ofradiation at the first holding device, a second array of planes ofradiation at the part and a third array of planes of radiation at thesecond holding device in the part-retaining position so that the holdingdevices and the part occlude their respective planes of radiation tocreate corresponding arrays of unobstructed planar portions of theplanes of radiation wherein each of the unobstructed planar portionscontains an amount of radiation which is representative of a respectiveexternal geometric dimension of one of the holding devices or the part,the head apparatus further including a plurality of receiver modules formeasuring the amount of radiation present in each of the unobstructedplanar portions to obtain holding device and part signals; a movablestage subsystem coupled to the head apparatus for translating the headapparatus relative to the holding devices and the part so that theplanes of radiation scan the holding devices and the part in thepart-retaining position; a signal processor for processing the holdingdevice and part signals to obtain data; and a data processor forprocessing the data to obtain the direct measurements, the dataprocessor determining the geometric dimensions related to the openingsbased on the direct measurements.
 13. The system as claimed in claim 12,wherein the holding devices are movable relative to each other betweenthe part-retaining position and a part-release position in which thedevices release the part for removal of the part from the station. 14.The system as claimed in claim 12, wherein the portion of each of theholding devices which extends into its respective opening has a slopedregion at which the portions seat on their respective openings at theirexterior surfaces and wherein each of the portions has a cross sectionalshape substantially identical to the shape of its respective opening atits exterior surface and wherein each sloped region defines a range ofsizes of its respective opening at its exterior surface that can beindirectly measured at the station.
 15. The system as claimed in claim14, wherein the portion of each of the devices is frustum-shaped andwherein each geometric dimension is distance across its respectiveopening at its exterior surfaces.
 16. The system as claimed in claim 12,wherein one of the geometric dimensions related to the openings isthickness of a wall which defines one of the openings at its exteriorsurface.
 17. The system as claimed in claim 14, wherein each of theholding devices includes a spring for spring-loading its respectiveportion so that the portions of the devices apply a variable clampingforce to the part.
 18. The system as claimed in claim 14, wherein eachof the sloped regions has a constant slope.
 19. The system as claimed inclaim 12, further comprising memory coupled to the data processor tostore data which represents height of the portion of the first holdingdevice wherein the portion of the first holding device has a distal endsurface which engages an inner end surface of its opening in thepart-retaining position and wherein the data processor processes thedata including the stored data to obtain depth of its opening in thepart.
 20. The system as claimed in claim 12, wherein the first holdingdevice includes first and second portions which extend into theirrespective opening in the part-retaining position, the first portionhaving a sloped region and a cross sectional shape substantiallyidentical to the shape of its opening at its exterior surface andwherein the sloped region defines a range of sizes of its opening at itsexterior surface that can be indirectly measured at the station andwherein the second portion has a distal end surface which engages aninner end surface of its opening in the part-retaining position.
 21. Thesystem as claimed in claim 20, wherein the first and second portions aremovable relative to each other and wherein the first holding devicefurther includes a spring for spring loading the first portion so thatthe first portion can apply a variable clamping force to the part. 22.The system as claimed in claim 12, wherein the part is an ammunitioncase.
 23. A system for indirectly measuring geometric dimensions relatedto openings in apertured end surfaces of an ammunition case based ondirect measurements of the case when fixtured at a measurement station,one end surface of the case being located at a mouth end of the case andthe other end surface being located at a primer end of the case, thesystem comprising: first and second holding devices for holding the casetherebetween in a case-retaining position in which the case is firmlyheld between the devices at the end surfaces of the case, wherein aportion of each of the holding devices extends into its respectiveopening in the case-retaining position; a head apparatus including aplurality of radiation sources for successively directing a first arrayof planes of radiation at the first holding device, a second array ofplanes of radiation at the case and a third array of planes of radiationat the second holding device in the case-retaining position so that theholding devices and the case occlude their respective planes ofradiation to create corresponding arrays of unobstructed planar portionsof the planes of radiation wherein each of the unobstructed planarportions contains an amount of radiation which is representative of arespective external geometric dimension of one of the holding devices orthe case, the head apparatus further including a plurality of receivermodules for measuring the amount of radiation present in each of theunobstructed planar portions to obtain holding device and case signals;a movable stage subsystem coupled to the head apparatus for translatingthe head apparatus relative to the holding devices and the case so thatthe planes of radiation scan the holding devices and the case in thecase-retaining position; a signal processor for processing the holdingdevice and case signals to obtain data; and a data processor forprocessing the data to obtain the direct measurements, the dataprocessor determining geometric dimensions related to the openings basedon the direct measurements.
 24. The system as claimed in claim 23,wherein the holding devices are movable relative to each other betweenthe case-retaining position and a case-release position in which thedevices release the case for removal of the case from the station. 25.The system as claimed in claim 23, wherein the portion of each of theholding devices which extends into its respective opening has a slopedregion at which the portions seat on their respective openings at theirexterior surfaces and wherein each of the portions was a cross sectionalshape substantially identical to the shape of its respective opening atits exterior surface and wherein each sloped region defines a range ofsizes of its respective opening at its exterior surface that can beindirectly measured at the station.
 26. The system as claimed in claim25, wherein the portion of each of the devices is frustum-shaped andwherein the geometric dimensions are inner diameter located at the mouthend and primer pocket diameter located at the primer end.
 27. The systemas claimed in claim 23, wherein one of the geometric dimensions relatedto the openings is thickness of a wall which defines the openings at themouth end.
 28. The system as claimed in claim 25, wherein each of theholding devices includes a spring for spring-loading its respectiveportion so that the portions of the devices apply a variable clampingforce to the case.
 29. The system as claimed in claim 25, wherein eachof the sloped regions has a constant slope.
 30. The system as claimed inclaim 23, further comprising memory coupled to the data processor tostore data which represents height of the portion of the first holdingdevice wherein the portion of the first holding device has a distal endsurface which engages an inner end surface of a primer pocket located atthe primer end in the case-retaining position and wherein the dataprocessor processes the data including the stored data to obtain primerpocket depth.
 31. The system as claimed in claim 23, wherein the firstholding device includes first and second portions which extend intotheir respective opening in the case-retaining position, the firstportion having a sloped region and a cross sectional shape substantiallyidentical to the shape of its opening at its end surface and wherein thesloped region defines a range of sizes of the opening at the primer endthat can be indirectly measured at the station and wherein the secondportion has a distal end surface which engages an inner end surface of aprimer pocket located at the primer end in the case-retaining position.32. The system as claimed in claim 31, wherein the first and secondportions are movable relative to each other and wherein the firstholding device further includes a spring for spring loading the firstportion so that the first portion can apply a variable clamping force tothe case.
 33. The system as claimed in claim 23, wherein the ammunitioncase is a cartridge case.
 34. The system as claimed in claim 1, whereineach of the receiver modules includes a camera.
 35. The system asclaimed in claim 34, wherein each of the cameras is a line-scan camera.36. The system as claimed in claim 12, wherein each of the receivermodules includes a camera.
 37. The system as claimed in claim 36,wherein each of the cameras is a line-scan camera.
 38. The system asclaimed in claim 23, wherein each of the receiver modules includes acamera.
 39. The system as claimed in claim 38, wherein each of thecameras is a line-scan camera.